id string | problem_id string | collection string | source string | source_title string | chapter int64 | chapter_title string | group string | difficulty string | problem string | solution string | final_answer string | final_answer_boxed string | answer_status string | verification_note string | page int64 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
dgn.core.1.001 | 1 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | A | Simplify the expression: $$\frac{\frac{1}{a}-\frac{1}{b+c}}{\frac{1}{a}+\frac{1}{b+c}} \cdot \left(1+\frac{b^2+c^2-a^2}{2bc}\right) : \frac{a-b-c}{abc}; \quad a=0.02, \; b=-11.05, \; c=1.07.$$ | Introduce the notation:
$$\frac{\frac{1}{a}-\frac{1}{b+c}}{\frac{1}{a}+\frac{1}{b+c}}=A;\;\left(1+\frac{b^2+c^2-a^2}{2bc}\right)=B;\;\frac{a-b-c}{abc}=C.$$
Then the given expression takes the form $A \cdot B : C$. In the expression $A$, the admissible values are $a \neq 0$, $b \neq 0$, $c \neq 0$, $b \neq -c$. Bringin... | 0.1 | \boxed{0.1} | verified | numeric(8 pts) | 8 | |
dgn.core.1.002 | 2 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | A | Simplify the expression: $\left(\frac{1}{t^2+3t+2}+\frac{2t}{t^2+4t+3}+\frac{1}{t^2+5t+6}\right)^2\frac{(t-3)^2+12t}{2}$. | To simplify the denominator, factor the quadratic trinomials:
$$t^2+3t+2=(t+1)(t+2);\qquad t^2+4t+3=(t+1)(t+3);\qquad t^2+5t+6=(t+2)(t+3).$$
Hence it is clear that for the fractions to exist we need $t\ne-1$, $t\ne-2$, and $t\ne-3$. Bringing the sum of the fractions to a common denominator, we obtain
$$\frac{1}{(t+1)(t... | 2 | \boxed{2} | verified | numeric(8 pts) | 8 | |
dgn.core.1.003 | 3 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | A | Simplify the expression: $\frac{(a-b)^2+ab}{(a+b)^2-ab}:\frac{a^5+b^5+a^2b^3+a^3b^2}{(a^3+b^3+a^2b+ab^2)(a^3-b^3)}$. | Let
$$A=\frac{(a-b)^2+ab}{(a+b)^2-ab};\quad
B=a^5+b^5+a^2b^3+a^3b^2;\quad
C=(a^3+b^3+a^2b+ab^2)(a^3-b^3).$$
Transform each of the expressions written above, using | no-gold | 8 | ||||
dgn.core.1.004 | 4 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | A | Simplify the expression: $\left(\frac{1+x+x^2}{2x+x^2}+2\frac{1-x+x^2}{2x-x^2}\right)^{-1}(5-2x^2)$; $x=\sqrt{3.92}$. | $0.04$.
The final answer is $\boxed{0.04}$. | 0.04 | \boxed{0.04} | suspected | numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wrong printed answer | 8 | |
dgn.core.1.005 | 5 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | A | Simplify the expression: $\frac{\left(m^2-\frac{1}{n^2}\right)^m\left(n+\frac{1}{m}\right)^{n-m}}{\left(n^2-\frac{1}{m^2}\right)^n\left(m-\frac{1}{n}\right)^{m-n}}$. | $(m/n)^{m+n}$.
The final answer is $\boxed{(m/n)^{m+n}}$. | (m/n)^{m+n} | \boxed{(m/n)^{m+n}} | verified | numeric(5 pts) | 8 | |
dgn.core.1.006 | 6 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | A | Simplify the expression: $\frac{\frac{a-b}{2a-b}-\frac{a^2+b^2+a}{2a^2+ab-b^2}}{(4b^4+4ab^2+a^2):(2b^2+a)}(b^2+b+ab+a)$. | $(b+1)/(b-2a)$.
The final answer is $\boxed{(b+1)/(b-2a)}$. | (b+1)/(b-2a) | \boxed{(b+1)/(b-2a)} | verified | numeric(8 pts) | 8 | |
dgn.core.1.007 | 7 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | A | Simplify the expression: $\frac{2(x^4+4x^2-12)+x^4+11x^2+30}{x^2+6}$. | $1+3x^2$.
The final answer is $\boxed{1+3x^2}$. | 1+3x^2 | \boxed{1+3x^2} | verified | numeric(8 pts) | 8 | |
dgn.core.1.008 | 8 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | A | Simplify the expression: $\frac{3a^2+2ax-x^2}{(3x+a)(a+x)}-2+10\frac{ax-3x^2}{a^2-9x^2}$. | $1$.
The final answer is $\boxed{1}$. | 1 | \boxed{1} | verified | numeric(8 pts) | 8 | |
dgn.core.1.009 | 9 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | A | Simplify the expression: $\left(\left(\frac{x}{y-x}\right)^{-2}-\frac{(x+y)^2-4xy}{x^2-xy}\right)^2\frac{x^4}{x^2y^2-y^4}$. | $(x-y)/(x+y)$.
The final answer is $\boxed{(x-y)/(x+y)}$. | (x-y)/(x+y) | \boxed{(x-y)/(x+y)} | verified | numeric(8 pts) | 8 | |
dgn.core.1.010 | 10 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | A | Simplify the expression: $\left(\left(\left(\frac{1}{a}+\frac{1}{b+c}\right):\left(\frac{1}{a}-\frac{1}{b+c}\right)\right):\left(1+\frac{b^2+c^2-a^2}{2bc}\right)\right);$ $a=1\frac{33}{40}$, $b=0.625$, $c=3.2$. | $1$.
The final answer is $\boxed{1}$. | 1 | \boxed{1} | verified | numeric(8 pts) | 8 | |
dgn.core.1.011 | 11 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(\left(\frac{x^2}{y^3}+\frac{1}{x}\right):\left(\frac{x}{y^2}-\frac{1}{y}+\frac{1}{x}\right)\right):\frac{(x-y)^2+4xy}{1+yx^{-1}}.$$ | $1/(xy)$.
The final answer is $\boxed{1/(xy)}$. | 1/(xy) | \boxed{1/(xy)} | verified | numeric(8 pts) | 9 | ||
dgn.core.1.012 | 12 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(\frac{3}{2x-y}-\frac{2}{2x+y}-\frac{1}{2x-5y}\right):\frac{y^2}{4x^2-y^2}.$$ | $24/(5y-2x)$.
The final answer is $\boxed{24/(5y-2x)}$. | 24/(5y-2x) | \boxed{24/(5y-2x)} | verified | numeric(8 pts) | 9 | ||
dgn.core.1.013 | 13 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Calculate: $$\left(x^2+2x-\frac{11x-2}{3x+1}\right):\left(x+1-\frac{2x^2+x+2}{3x+1}\right);\quad x=7,\;(3).$$ | $20$.
The final answer is $\boxed{20}$. | 20 | \boxed{20} | unverifiable | no '=' | 9 | ||
dgn.core.1.014 | 14 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(6a^2+5a-1+\frac{a+4}{a+1}\right):\left(3a-2+\frac{3}{a+1}\right).$$ | $2a+3$.
The final answer is $\boxed{2a+3}$. | 2a+3 | \boxed{2a+3} | verified | numeric(8 pts) | 9 | ||
dgn.core.1.015 | 15 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{x^{-6}-64}{4+2x^{-1}+x^{-2}}:\frac{x^{2}}{4-\frac{4}{x}+\frac{1}{x^{2}}}=\frac{4x^{2}(2x+1)}{1-2x}.$$ | $1+2x$.
The final answer is $\boxed{1+2x}$. | 1+2x | \boxed{1+2x} | unverifiable | unparsed question | 9 | ||
dgn.core.1.016 | 16 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Calculate: $$\frac{(2p-q)^2+2q^2-3pq}{2p^{-1}+q^2}:\frac{4p^2-3pq}{2+pq^2};\quad p=0.78,\;q=7/25.$$ | $0.5$.
The final answer is $\boxed{0.5}$. | 0.5 | \boxed{0.5} | unverifiable | no '=' | 9 | ||
dgn.core.1.017 | 17 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{2b+a-\frac{4a^2-b^2}{a}}{b^3+2ab^2-3a^2b}\cdot\frac{a^3b-2a^2b^2+ab^3}{a^2-b^2}.$$ | $(a-b)/(a+b)$.
The final answer is $\boxed{(a-b)/(a+b)}$. | (a-b)/(a+b) | \boxed{(a-b)/(a+b)} | verified | numeric(8 pts) | 9 | ||
dgn.core.1.018 | 18 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Calculate: $$\frac{\left(\frac{1}{a}+\frac{1}{b}-\frac{2c}{ab}\right)(a+b+2c)}{\frac{1}{a^2}+\frac{1}{b^2}+\frac{2}{ab}-\frac{4c^2}{a^2b^2}};\quad a=7.4,\;b=\frac{5}{37}.$$ | $1$.
The final answer is $\boxed{1}$. | 1 | \boxed{1} | unverifiable | no '=' | 9 | ||
dgn.core.1.020 | 20 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{1}{b(abc+a+c)}-\frac{1}{a+\frac{1}{b+\frac{1}{c}}}:\frac{1}{a+\frac{1}{b}}.$$ | $-1$.
The final answer is $\boxed{-1}$. | -1 | \boxed{-1} | verified | numeric(8 pts) | 9 | ||
dgn.core.1.022 | 22 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Calculate: $$\left(\frac{2-b}{b-1}+2\frac{a-1}{a-2}\right):\left(b\frac{a-1}{b-1}+a\frac{2-b}{a-2}\right);\quad a=\sqrt{2}+0.8,\;b=\sqrt{2}-0.2.$$ | $1$.
The final answer is $\boxed{1}$. | 1 | \boxed{1} | unverifiable | no '=' | 9 | ||
dgn.core.1.023 | 23 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{1+(a+x)^{-1}}{1-(a+x)^{-1}}\cdot\left(1-\frac{1-(a^2+x^2)}{2ax}\right);\quad x=\frac{1}{a-1}.$$ | $0.5a^3(a-1)$.
The final answer is $\boxed{0.5a^3(a-1)}$. | 0.5a^3(a-1) | \boxed{0.5a^3(a-1)} | suspected | the source answer is contradicted at a=3: the expression is exactly 27/4, the source answer gives 27 - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wrong printed answer | 9 | ||
dgn.core.1.024 | 24 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Calculate: $$\left(\frac{a}{b}+\frac{b}{a}+2\right)\left(\frac{a+b}{2a}-\frac{b}{a+b}\right):\left(\left(a+2b+\frac{b^2}{a}\right)\times\left(\frac{a}{a+b}+\frac{b}{a-b}\right)\right);\quad a=0.75,\;b=4/3.$$ | $-7/24$.
The final answer is $\boxed{-7/24}$. | -7/24 | \boxed{-7/24} | unverifiable | no '=' | 9 | ||
dgn.core.1.025 | 25 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{\left(ab^{-1}+a^{-1}b+1\right)\left(a^{-1}-b^{-1}\right)^2}{a^2b^{-2}+a^{-2}b^2-\left(ab^{-1}+a^{-1}b\right)}.$$ | $1/(ab)$.
The final answer is $\boxed{1/(ab)}$. | 1/(ab) | \boxed{1/(ab)} | verified | numeric(8 pts) | 10 | ||
dgn.core.1.026 | 26 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(\sqrt{\sqrt{m}-\sqrt{\frac{m^2-9}{m}}}+\sqrt{\sqrt{m}+\sqrt{\frac{m^2-9}{m}}}\right)^2\sqrt[4]{\frac{m^2}{4}}.$$ | Using formula (1.9) for the radical of the first factor of the given expression, we obtain
$$\sqrt{m}-\sqrt{\frac{m^2-9}{m}}+2\sqrt{m}-\frac{m^2-9}{m}+\sqrt{m}+\sqrt{\frac{m^2-9}{m}}=$$
$$=2\sqrt{m}+\frac{2\sqrt{m^2}-m^2+9}{\sqrt{m}}=\frac{2(m+3)}{\sqrt{m}}.$$
Further, we find
$$\frac{2(m+3)}{\sqrt{m}}\cdot\sqrt{\frac{... | \sqrt{2}\,(m+3) | \boxed{\sqrt{2}\,(m+3)} | verified | numeric(8 pts) | 10 | ||
dgn.core.1.027 | 27 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{9b^{4/3}-\frac{a^{3/2}}{b^2}}{\sqrt{a^{3/2}b^{-2}+6a^{3/4}b^{-1/3}+9b^{4/3}}\cdot\frac{b^2}{a^{3/4}-3b^{5/3}}};\quad b=4.$$ | Using formulas (1.7), (1.8), (1.9), we obtain
$$\frac{(3b^{2/3})^2-(a^{3/4}b^{-1})^2}{\sqrt{(a^{3/4}b^{-1})^2+2\cdot 3a^{3/4}b^{-1/3}+(3b^{2/3})^2}}=$$
The final answer is $\boxed{-4}$. | -4 | \boxed{-4} | suspected | the source answer is contradicted at a=1, b=4: the expression is exactly (-27649 + 24*2**(1/3) + 576*2**(2/3))/(64*sqrt(1 + 48*2**(1/3) + 576*2**(2/3))), the source answer gives -4 - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wrong printed answer | 10 | ||
dgn.core.1.028 | 28 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{\left(\sqrt[3]{(r^2+4)\sqrt{1+\frac{4}{r^2}}}-\sqrt[3]{(r^2-4)\sqrt{1-\frac{4}{r^2}}}\right)^2}{r^2-\sqrt{r^4-16}}.$$ | Using formulas (1.20) and (1.23), we transform the difference in the numerator: $\sqrt[3]{\frac{(r^2+4)^{3/2}}{|r|}}-\sqrt[3]{\frac{(r^2-4)^{3/2}}{|r|}}=\frac{\sqrt{r^2+4}-\sqrt{r^2-4}}{\sqrt[3]{|r|}}$. Consequently,
The final answer is $\boxed{\frac{2}{\sqrt[3]{r^2}}\left(r^2-\sqrt{r^4-16}\right)}$. | \frac{2}{\sqrt[3]{r^2}}\left(r^2-\sqrt{r^4-16}\right) | \boxed{\frac{2}{\sqrt[3]{r^2}}\left(r^2-\sqrt{r^4-16}\right)} | suspected | the source answer is contradicted at r=2: the expression is exactly 2**(1/3), the source answer gives 4*2**(1/3) - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wrong printed answer | 10 | ||
dgn.core.1.030 | 30 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(\frac{\sqrt{x-a}}{\sqrt{x+a}+\sqrt{x-a}}+\frac{x-a}{\sqrt{x^2-a^2}-x+a}\right):\sqrt{\frac{x^2}{a^2}-1};\quad x>a>0.$$ | Here, the only admissible values for the expression are the values $x>a$, where $a>0$. Using the equality $x-a=\sqrt{x-a}\cdot\sqrt{x-a}$ and bringing the fractions in the parentheses to a common denominator, we obtain
$$\frac{\sqrt{x-a}}{\sqrt{x+a}+\sqrt{x-a}}
+
\frac{\sqrt{x-a}\cdot\sqrt{x-a}}{\sqrt{x-a}\left(\sqrt{x... | 1 | \boxed{1} | verified | numeric(3 pts) | 10 | ||
dgn.core.1.031 | 31 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(\sqrt[3]{\left(\dfrac{1}{2}\right)^{-3}-t^3}+\sqrt[3]{\dfrac{t^5+2t^4+4t^3}{4-4t+t^2}}\right):\left(\dfrac{1}{\sqrt{2}-\sqrt{t}}+\dfrac{1}{\sqrt{t}+\sqrt{2}}\right).$$ | Applying formulas (1.7), (1.9), (1.10), (1.14) and (1.16), in the first parenthesis we obtain
$$\sqrt[3]{2^3-t^3}+\sqrt[3]{\frac{t^3(t^2+2t+4)}{(2-t)^2}}=\sqrt[3]{(2-t)(4+2t+t^2)}+$$
$$+t\sqrt[3]{\frac{t^2+2t+4}{(2-t)^2}}=\sqrt[3]{4+2t+t^2}\left(\sqrt[3]{2-t}+\frac{t}{\sqrt[3]{(2-t)^2}}\right)=$$
$$=\sqrt[3]{4+2t+t^2}\... | \frac{\sqrt[3]{8-t^3}}{\sqrt{2}} | \boxed{\frac{\sqrt[3]{8-t^3}}{\sqrt{2}}} | verified | third read of the page corrected the row | 10 | ||
dgn.core.1.032 | 32 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{\sqrt{x}+1}{x\sqrt{x}+x+\sqrt{x}}:\frac{1}{x^2-\sqrt{x}}.$$ | $x-1$.
The final answer is $\boxed{x-1}$. | x-1 | \boxed{x-1} | verified | numeric(8 pts) | 10 | ||
dgn.core.1.033 | 33 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(\left(\sqrt[4]{p}-\sqrt[4]{q}\right)^{-2}+\left(\sqrt[4]{p}+\sqrt[4]{q}\right)^{-2}\right):\frac{\sqrt{p}+\sqrt{q}}{p-q}.$$ | $2\left(\sqrt{p}+\sqrt{q}\right)^2/(p-q)$.
The final answer is $\boxed{2\left(\sqrt{p}+\sqrt{q}\right)^2/(p-q)}$. | 2\left(\sqrt{p}+\sqrt{q}\right)^2/(p-q) | \boxed{2\left(\sqrt{p}+\sqrt{q}\right)^2/(p-q)} | verified | numeric(5 pts) | 10 | ||
dgn.core.1.034 | 34 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{\left(\sqrt{a^2+a\sqrt{a^2-b^2}}-\sqrt{a^2-a\sqrt{a^2-b^2}}\right)^2}{2\sqrt{a^3b}}:\left(\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}-2\right);\quad a>b>0.$$ | $\left(\sqrt{a}+\sqrt{b}\right)^2/(a-b)$.
The final answer is $\boxed{\left(\sqrt{a}+\sqrt{b}\right)^2/(a-b)}$. | \left(\sqrt{a}+\sqrt{b}\right)^2/(a-b) | \boxed{\left(\sqrt{a}+\sqrt{b}\right)^2/(a-b)} | verified | numeric(2 pts) | 10 | ||
dgn.core.1.036 | 36 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{2x^{-1/3}}{x^{2/3}-3x^{-1/3}}-\frac{x^{2/3}}{x^{5/3}-x^{2/3}}-\frac{x+1}{x^2-4x+3}.$$ | $0$.
The final answer is $\boxed{0}$. | 0 | \boxed{0} | verified | numeric(8 pts) | 10 | ||
dgn.core.1.037 | 37 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{(\sqrt{a}+\sqrt{b})^2-4b}{(a-b):\left(\sqrt{\frac{1}{b}}+3\sqrt{\frac{1}{a}}\right):\frac{a+9b+6\sqrt{ab}}{\frac{1}{\sqrt{b}}+\frac{1}{\sqrt{a}}}.$$ | $1/(ab)$.
The final answer is $\boxed{1/(ab)}$. | 1/(ab) | \boxed{1/(ab)} | unverifiable | unparsed question | 10 | ||
dgn.core.1.038 | 38 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{\left(\sqrt[4]{m}+\sqrt[4]{n}\right)^2+\left(\sqrt[4]{m}-\sqrt[4]{n}\right)^2}{2(m-n)}:\frac{1}{\sqrt{m^3}-\sqrt{n^3}}-3\sqrt{mn}.$$ | $\left(\sqrt{m}-\sqrt{n}\right)^2$.
The final answer is $\boxed{\left(\sqrt{m}-\sqrt{n}\right)^2}$. | \left(\sqrt{m}-\sqrt{n}\right)^2 | \boxed{\left(\sqrt{m}-\sqrt{n}\right)^2} | verified | numeric(5 pts) | 11 | ||
dgn.core.1.039 | 39 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(\left(\frac{2^{3/2}+27y^{3/5}}{\sqrt{2}+3\sqrt[5]{y}}+3\sqrt[10]{32y^2-2}\,3^{-2}\right)\right)^5.$$ | $y^2$.
The final answer is $\boxed{y^2}$. | y^2 | \boxed{y^2} | suspected | the source answer is contradicted at y=1: the expression is exactly (30**(1/10)*(sqrt(2) + 3) + 6*sqrt(2) + 81)**5/(243*(sqrt(2) + 3)**5), the source answer gives 1 - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wrong printed answer | 11 | ||
dgn.core.1.041 | 41 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$t\cdot\frac{1+\frac{2}{\sqrt{t}+4}}{2-\sqrt{t}+4}+\sqrt{t}+4+\frac{4}{\sqrt{t}+4}.$$ | $-4$.
The final answer is $\boxed{-4}$. | -4 | \boxed{-4} | suspected | the source answer is contradicted at t=1: the expression is exactly 152/25, the source answer gives -4 - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wrong printed answer | 11 | ||
dgn.core.1.042 | 42 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(\frac{1+\sqrt{x}}{\sqrt{1+x}}-\frac{\sqrt{1+x}}{1+\sqrt{x}}\right)^2-\left(\frac{1-\sqrt{x}}{\sqrt{1+x}}-\frac{\sqrt{1+x}}{1-\sqrt{x}}\right)^2.$$ | $\frac{16x\sqrt{x}}{(1-x^2)(x-1)}$.
The final answer is $\boxed{\frac{16x\sqrt{x}}{(1-x^2)(x-1)}}$. | \frac{16x\sqrt{x}}{(1-x^2)(x-1)} | \boxed{\frac{16x\sqrt{x}}{(1-x^2)(x-1)}} | verified | numeric(8 pts) | 11 | ||
dgn.core.1.043 | 43 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{x-1}{x+x^{1/2}+1}:\frac{x^{0,5}+1}{x^{1,5}-1}+\frac{2}{x^{-0,5}}.$$ | $x+1$.
The final answer is $\boxed{x+1}$. | x+1 | \boxed{x+1} | verified | numeric(8 pts) | 11 | ||
dgn.core.1.045 | 45 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{x-y}{x^{3/4}+x^{1/2}y^{1/4}}\cdot\frac{x^{1/2}y^{1/4}+x^{1/4}y^{1/2}}{x^{1/2}+y^{1/2}}\cdot\frac{x^{1/4}y^{-1/4}}{x^{1/2}-2x^{1/4}y^{1/4}+y^{1/2}}.$$ | $\frac{\sqrt[4]{x}+\sqrt[4]{y}}{\sqrt[4]{x}-\sqrt[4]{y}}$.
The final answer is $\boxed{\frac{\sqrt[4]{x}+\sqrt[4]{y}}{\sqrt[4]{x}-\sqrt[4]{y}}}$. | \frac{\sqrt[4]{x}+\sqrt[4]{y}}{\sqrt[4]{x}-\sqrt[4]{y}} | \boxed{\frac{\sqrt[4]{x}+\sqrt[4]{y}}{\sqrt[4]{x}-\sqrt[4]{y}}} | verified | numeric(5 pts) | 11 | ||
dgn.core.1.046 | 46 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\sqrt[n]{y^{\frac{2n}{m-n}}}:\sqrt[m]{y^{\frac{(m-n)^2+4mn}{m^2-n^2}}}.$$ | $\sqrt[m]{y}$.
The final answer is $\boxed{\sqrt[m]{y}}$. | \sqrt[m]{y} | \boxed{\sqrt[m]{y}} | verified | numeric(8 pts) | 11 | ||
dgn.core.1.047 | 47 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(\frac{\left(z^{2/p}+z^{2/q}\right)^2-4z^{2/p+2/q}}{\left(z^{1/p}-z^{1/q}\right)^2+4z^{1/p+1/q}}\right)^{1/2}.$$ | $\left|z^{1/p}-z^{1/q}\right|$.
The final answer is $\boxed{\left|z^{1/p}-z^{1/q}\right|}$. | \left|z^{1/p}-z^{1/q}\right| | \boxed{\left|z^{1/p}-z^{1/q}\right|} | verified | numeric(8 pts) | 11 | ||
dgn.core.1.048 | 48 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{x-1}{x^{3/4}+x^{1/2}}\cdot\frac{x^{1/2}+x^{1/4}}{x^{1/2}+1}\cdot x^{1/4}+1.$$ | $\sqrt{x}$.
The final answer is $\boxed{\sqrt{x}}$. | \sqrt{x} | \boxed{\sqrt{x}} | verified | numeric(8 pts) | 11 | ||
dgn.core.1.049 | 49 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Calculate at $x=64$: $$\frac{(x^2-y^2)(\sqrt[3]{x}+\sqrt[3]{y})}{\sqrt[3]{x^6}+\sqrt[3]{x^2y^2}-\sqrt[3]{x^2y^2}-\sqrt[3]{y^6}}-\left(\sqrt[3]{xy}+\sqrt[3]{y^2}\right).$$ | $16$.
The final answer is $\boxed{16}$. | 16 | \boxed{16} | unverifiable | no '=' | 11 | ||
dgn.core.1.050 | 50 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\sqrt{\frac{2a}{(1+a)^3\sqrt{1+a}}}\cdot\sqrt[3]{\frac{4+8/a+4/a^2}{\sqrt{2}}}.$$ | $2\sqrt[6]{a^5/a}$.
The final answer is $\boxed{2\sqrt[6]{a^5/a}}$. | 2\sqrt[6]{a^5/a} | \boxed{2\sqrt[6]{a^5/a}} | suspected | the source answer is contradicted at a=1: the expression is exactly 2**(11/12)/2, the source answer gives 2 - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wrong printed answer | 11 | ||
dgn.core.1.051 | 51 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{4x\left(x+\sqrt{x^2-1}\right)^2}{\left(x+\sqrt{x^2-1}\right)^4-1}.$$ | $1/\sqrt{x^2-1}$.
The final answer is $\boxed{1/\sqrt{x^2-1}}$. | 1/\sqrt{x^2-1} | \boxed{1/\sqrt{x^2-1}} | verified | numeric(8 pts) | 12 | ||
dgn.core.1.052 | 52 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{\sqrt{(x+2)^2-8x}}{\sqrt{x-2}\sqrt{x}}.$$ | $-\sqrt{x}$ for $x\in(0,2)$; $\sqrt{x}$ for $x\in(2,\infty)$.
The final answer is $\boxed{-\sqrt{x} for x\in(0,2); \sqrt{x} for x\in(2,\infty)}$. | -\sqrt{x} for x\in(0,2); \sqrt{x} for x\in(2,\infty) | \boxed{-\sqrt{x} for x\in(0,2); \sqrt{x} for x\in(2,\infty)} | unverifiable | unparsed answer | 12 | ||
dgn.core.1.054 | 54 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\sqrt[6]{4x\left(11+4\sqrt{6}\right)}\cdot\sqrt[3]{4}\sqrt{2x}-2\sqrt{3x}.$$ | $\sqrt[3]{20x}$.
The final answer is $\boxed{\sqrt[3]{20x}}$. | \sqrt[3]{20x} | \boxed{\sqrt[3]{20x}} | suspected | the source answer is contradicted at x=1: the expression is exactly -2*sqrt(3) + 2*sqrt(2)*(4*sqrt(6) + 11)**(1/6), the source answer gives 2**(2/3)*5**(1/3) - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wrong printed answer | 12 | ||
dgn.core.1.056 | 56 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{\left(\sqrt[5]{a^{4/3}}\right)^{3/2}\cdot\left(\sqrt{a^3}\sqrt{a^2b}\right)^4}{\left(\sqrt[5]{a^4}\right)^3\cdot\left(\sqrt[4]{a}\sqrt{b}\right)^6}.$$ | $1/\sqrt[12]{a^2b}$.
The final answer is $\boxed{1/\sqrt[12]{a^2b}}$. | 1/\sqrt[12]{a^2b} | \boxed{1/\sqrt[12]{a^2b}} | suspected | the source answer is contradicted at a=1, b=2: the expression is exactly 1/2, the source answer gives 2**(11/12)/2 - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wrong printed answer | 12 | ||
dgn.core.1.057 | 57 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{\sqrt[3]{x+\sqrt{2-x^2}\cdot\sqrt[6]{1-x\sqrt{2-x^2}}}{\sqrt[3]{1-x^2}}.$$ | $\pm\sqrt[6]{2}$.
The final answer is $\boxed{\pm\sqrt[6]{2}}$. | \pm\sqrt[6]{2} | \boxed{\pm\sqrt[6]{2}} | unverifiable | unparsed question | 12 | ||
dgn.core.1.058 | 58 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{\frac{x\left(x^2-a^2\right)^{-1/2}+1}{a\left(x-a\right)^{-1/2}+\left(x-a\right)^{1/2}}:\frac{a^2\sqrt{x}+a}{x-\left(x^2-a^2\right)^{1/2}}+\frac{1}{x^2-ax}.$$ | $2/(x^2-a^2)$.
The final answer is $\boxed{2/(x^2-a^2)}$. | 2/(x^2-a^2) | \boxed{2/(x^2-a^2)} | unverifiable | unparsed question | 12 | ||
dgn.core.1.059 | 59 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\sqrt{\sqrt{\frac{2}{a}}+\frac{a}{\sqrt{2}}+2}-\frac{a^2\sqrt[4]{2}-2\sqrt{a}}{a\sqrt{2a}-\sqrt[4]{8a^4}}.$$ | $-1$.
The final answer is $\boxed{-1}$. | -1 | \boxed{-1} | suspected | the source answer is contradicted at a=2: the expression is exactly -2**(1/4) - 1 - 2**(3/4)/2 + sqrt(sqrt(2) + 3), the source answer gives -1 - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wrong printed answer | 12 | ||
dgn.core.1.060 | 60 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(\frac{\sqrt[4]{a^3}-1}{\sqrt[4]{a}-1}+\sqrt[4]{a}\right)^{1/2}\left(\frac{\sqrt[4]{a^3}+1}{\sqrt[4]{a}+1}-\sqrt{a}\right)\left(a-\sqrt{a^3}\right)^{-1}.$$ | $1/a$.
The final answer is $\boxed{1/a}$. | 1/a | \boxed{1/a} | verified | numeric(8 pts) | 12 | ||
dgn.core.1.061 | 61 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{\sqrt{x^3}+\sqrt{xy^2}-\sqrt{x^2y}-\sqrt{y^3}}{\sqrt[4]{y^3}+\sqrt[4]{x^4y}-\sqrt[4]{xy^4}-\sqrt[4]{x^3}}.$$ | $-(\sqrt[4]{x}-\sqrt[4]{y})$.
The final answer is $\boxed{-(\sqrt[4]{x}-\sqrt[4]{y})}$. | -(\sqrt[4]{x}-\sqrt[4]{y}) | \boxed{-(\sqrt[4]{x}-\sqrt[4]{y})} | suspected | the source answer is contradicted at x=1, y=2: the expression is exactly 3*(1 - sqrt(2))/(-3 + 2**(1/4) + 2**(3/4)), the source answer gives -1 + 2**(1/4) - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wrong printed answer | 12 | ||
dgn.core.1.062 | 62 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{a^{1/2}+ab^{-1}}{a^{-1/3}-a^{-1/6}b^{-1/3}+b^{-2/3}}-\frac{a}{\sqrt[3]{b}}.$$ | $a^{5/6}$.
The final answer is $\boxed{a^{5/6}}$. | a^{5/6} | \boxed{a^{5/6}} | verified | numeric(5 pts) | 12 | ||
dgn.core.1.063 | 63 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{a^{7/3}-2a^{5/3}b^{2/3}+ab^{4/3}}{a^{5/3}-a^{4/3}b^{1/3}-ab^{2/3}+a^{2/3}b^{2/3}}:a^{1/3}.$$ | $a^{1/3}+b^{1/3}$.
The final answer is $\boxed{a^{1/3}+b^{1/3}}$. | a^{1/3}+b^{1/3} | \boxed{a^{1/3}+b^{1/3}} | suspected | the source answer is contradicted at a=1, b=2: the expression is exactly (-2*2**(2/3) + 1 + 2*2**(1/3))/(1 - 2**(1/3)), the source answer gives 1 + 2**(1/3) - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wrong printed answer | 12 | ||
dgn.core.1.064 | 64 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{\left(a^2-b^2\right)\left(\sqrt[3]{a}-\sqrt[3]{b}\right)}{\sqrt[3]{a^4}+\sqrt[3]{ab^3}-\sqrt[3]{a^3b}-\sqrt[3]{b^4}}.$$ | $a-b$.
The final answer is $\boxed{a-b}$. | a-b | \boxed{a-b} | verified | numeric(5 pts) | 12 | ||
dgn.core.1.065 | 65 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{(m-1)\sqrt{m}-(n-1)\sqrt{n}}{\sqrt{m^3n+mn+m^2}-m}.$$ | $(\sqrt{m}-\sqrt{n})/m$.
The final answer is $\boxed{(\sqrt{m}-\sqrt{n})/m}$. | (\sqrt{m}-\sqrt{n})/m | \boxed{(\sqrt{m}-\sqrt{n})/m} | suspected | the source answer is contradicted at m=1, n=2: the expression is exactly -sqrt(sqrt(5)/4 + 3/4), the source answer gives 1 - sqrt(2) - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wrong printed answer | 12 | ||
dgn.core.1.066 | 66 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{\sqrt[3]{ab}\left(\sqrt[3]{b^2}-\sqrt[3]{a^2}\right)+\sqrt[3]{a^4}-\sqrt[3]{b^4}}{\sqrt[3]{a^4}+\sqrt[3]{a^2b^2}-\sqrt[3]{a^3b}}\cdot\sqrt[3]{a^2}.$$ | $\sqrt[3]{a^2}-\sqrt[3]{b^2}$.
The final answer is $\boxed{\sqrt[3]{a^2}-\sqrt[3]{b^2}}$. | \sqrt[3]{a^2}-\sqrt[3]{b^2} | \boxed{\sqrt[3]{a^2}-\sqrt[3]{b^2}} | verified | numeric(5 pts) | 13 | ||
dgn.core.1.067 | 67 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{\sqrt{5}-2\sqrt{6}}{\left(\sqrt[4]{3}+\sqrt[4]{2}\right)\left(\sqrt[4]{3}-\sqrt[4]{2}\right)}.$$ | $1$.
The final answer is $\boxed{1}$. | 1 | \boxed{1} | suspected | the source answer is contradicted at : the expression is exactly (-sqrt(5) + 2*sqrt(6))/(-sqrt(3) + sqrt(2)), the source answer gives 1 - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wrong printed answer | 13 | ||
dgn.core.1.068 | 68 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{\left(a^{1/m}-a^{1/n}\right)^2+4a^{(m+n)/(mn)}}{\left(a^{2/m}-a^{2/n}\right)^m\sqrt[m]{a^{m+1}}+\sqrt[n]{a^{n+1}}}.$$ | $\frac{1}{a\left(\sqrt[m]{a}-\sqrt[n]{a}\right)}$.
The final answer is $\boxed{\frac{1}{a\left(\sqrt[m]{a}-\sqrt[n]{a}\right)}}$. | \frac{1}{a\left(\sqrt[m]{a}-\sqrt[n]{a}\right)} | \boxed{\frac{1}{a\left(\sqrt[m]{a}-\sqrt[n]{a}\right)}} | suspected | the source answer is contradicted at a=2, m=1, n=2: the expression is exactly 5*sqrt(2)/14 + 4/7, the source answer gives sqrt(2)/4 + 1/2 - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wrong printed answer | 13 | ||
dgn.core.1.069 | 69 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{\sqrt[3]{12}}{\sqrt{45}-4\sqrt{3}}+5\sqrt{2,4}\left(\sqrt{15}+3\right).$$ | $6$.
The final answer is $\boxed{6}$. | 6 | \boxed{6} | suspected | the source answer is contradicted at : the expression is exactly (-18*sqrt(5) - 12**(1/3) + 30*sqrt(3))/(-3*sqrt(5) + 4*sqrt(3)), the source answer gives 6 - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wrong printed answer | 13 | ||
dgn.core.1.070 | 70 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(\frac{t\sqrt{t+2}}{\sqrt{t-2}}-\frac{2\sqrt{t-2}}{\sqrt{t+2}}-\frac{4t}{\sqrt{t^2-4}}\right)^{1/2}:\sqrt[4]{t^2-4}.$$ | $\sqrt{t^2-4}/(t+2)$.
The final answer is $\boxed{\sqrt{t^2-4}/(t+2)}$. | \sqrt{t^2-4}/(t+2) | \boxed{\sqrt{t^2-4}/(t+2)} | verified | numeric(8 pts) | 13 | ||
dgn.core.1.071 | 71 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(\frac{a\sqrt{a}+b\sqrt{b}}{\sqrt{a}+\sqrt{b}}-\sqrt{ab}\right)\left(\frac{\sqrt{a}+\sqrt{b}}{a-b}\right)^2.$$ | $1$.
The final answer is $\boxed{1}$. | 1 | \boxed{1} | verified | numeric(5 pts) | 13 | ||
dgn.core.1.073 | 73 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{\sqrt{3}(a-b^2)+\sqrt{3}\,b^3\sqrt{8b^3}}{\sqrt{2}(a-b^2)^2+(2b\sqrt{2a})^2}\cdot\frac{\sqrt{2a}-\sqrt{2c}}{\sqrt[3]{a}-\sqrt[3]{c}}.$$ | $-\sqrt{ac}$.
The final answer is $\boxed{-\sqrt{ac}}$. | -\sqrt{ac} | \boxed{-\sqrt{ac}} | suspected | the source answer is contradicted at a=1, b=1, c=2: the expression is exactly 2**(190/213)*3**(308/639)*5**(407/639)*7**(569/639)/36, the source answer gives -sqrt(2) - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wrong printed answer | 13 | ||
dgn.core.1.074 | 74 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(\sqrt{1-x^2}+1\right):\left(\frac{1}{\sqrt{1+x}}+\sqrt{1-x}\right).$$ | $\sqrt{1+x}$.
The final answer is $\boxed{\sqrt{1+x}}$. | \sqrt{1+x} | \boxed{\sqrt{1+x}} | verified | numeric(8 pts) | 13 | ||
dgn.core.1.075 | 75 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{8-n}{2+\sqrt[3]{n}}:\left(2+\frac{\sqrt[3]{n^2}}{2+\sqrt[3]{n}}\right)-\left(\sqrt[3]{n}+\frac{2\sqrt[3]{n}}{\sqrt[3]{n}-2}\right)\frac{4-\sqrt[3]{n^2}}{\sqrt[3]{n^2}+2\sqrt[3]{n}}.$$ | $2$.
The final answer is $\boxed{2}$. | 2 | \boxed{2} | verified | numeric(8 pts) | 13 | ||
dgn.core.1.076 | 76 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{(a-b)^3\left(\sqrt{a}+\sqrt{b}\right)^{-3}+2a\sqrt{a}+b\sqrt{b}}{a\sqrt{a}+b\sqrt{b}}+\frac{3\left(\sqrt{ab}-b\right)}{a-b}.$$ | $3$.
The final answer is $\boxed{3}$. | 3 | \boxed{3} | verified | numeric(5 pts) | 13 | ||
dgn.core.1.077 | 77 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{x^{1/6}-y^{1/6}}{x^{1/2}+x^{1/3}y^{1/6}}\cdot\frac{\left(x^{1/3}+y^{1/3}\right)^2-4\sqrt[3]{xy}}{x^{5/6}y^{1/3}-x^{1/2}y^{2/3}}+2x^{-2/3}y^{-1/6}.$$ | $\frac{x^{1/3}+y^{1/3}}{\sqrt[6]{x^3y^2}}$.
The final answer is $\boxed{\frac{x^{1/3}+y^{1/3}}{\sqrt[6]{x^3y^2}}}$. | \frac{x^{1/3}+y^{1/3}}{\sqrt[6]{x^3y^2}} | \boxed{\frac{x^{1/3}+y^{1/3}}{\sqrt[6]{x^3y^2}}} | suspected | the source answer is contradicted at x=2, y=1: the expression is exactly (-2**(1/6) - 1 + 2**(2/3) + 2**(5/6))/(-2 - 2**(5/6) + 2*2**(1/6) + 2*2**(1/3)), the source answer gives sqrt(2)/2 + 2**(5/6)/2 - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wr... | 13 | ||
dgn.core.1.078 | 78 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(x\sqrt[3]{\frac{x-1}{(x+1)^2}}+\frac{x-1}{\sqrt[3]{(x^2-1)^2}}\right)^{-3/5}:(x^2-1)^{4/5}.$$ | $1/(x^2-1)$.
The final answer is $\boxed{1/(x^2-1)}$. | 1/(x^2-1) | \boxed{1/(x^2-1)} | verified | numeric(8 pts) | 13 | ||
dgn.core.1.079 | 79 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(\frac{\sqrt{3}+1}{1+\sqrt{3}+\sqrt{t}}+\frac{\sqrt{3}-1}{1-\sqrt{3}+\sqrt{t}}\right)\left(\sqrt{t}-\frac{2}{\sqrt{t}}+2\right).$$ | $2\sqrt{3}$.
The final answer is $\boxed{2\sqrt{3}}$. | 2\sqrt{3} | \boxed{2\sqrt{3}} | verified | numeric(8 pts) | 13 | ||
dgn.core.1.080 | 80 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{m^{4/3}-27m^{1/3}\cdot n}{m^{2/3}+3\sqrt[3]{mn}+9n^{2/3}}:\left(1-3\sqrt[3]{\frac{n}{m}}\right)-3\sqrt{m^2}.$$ | $0$.
The final answer is $\boxed{0}$. | 0 | \boxed{0} | suspected | the source answer is contradicted at m=1, n=1: the expression is exactly -2, the source answer gives 0 - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wrong printed answer | 13 | ||
dgn.core.1.081 | 81 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$z^{\frac{p-3}{p^2+3p}}:z^{\frac{12}{9-p^2}}:z^{\frac{3}{3p-p^2}}.$$ | $z^{1/(p-3)}$.
The final answer is $\boxed{z^{1/(p-3)}}$. | z^{1/(p-3)} | \boxed{z^{1/(p-3)}} | disputed | third read of the page disagrees with the bulk transcription (statement_corrected_but_still_wrong) | 14 | ||
dgn.core.1.082 | 82 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\sqrt{\frac{x}{x-a^2}}:\left(\frac{\sqrt{x}-\sqrt{x-a^2}}{\sqrt{x}+\sqrt{x-a^2}}-\frac{\sqrt{x}+\sqrt{x-a^2}}{\sqrt{x}-\sqrt{x-a^2}}\right).$$ | $\frac{a^2}{4(a^2-x)}$.
The final answer is $\boxed{\frac{a^2}{4(a^2-x)}}$. | \frac{a^2}{4(a^2-x)} | \boxed{\frac{a^2}{4(a^2-x)}} | verified | numeric(8 pts) | 14 | ||
dgn.core.1.083 | 83 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{\sqrt{\frac{abc+4}{a}}+4\sqrt{\frac{bc}{a}}}{\sqrt{abc+2}};\quad a=0.04.$$ | $5$.
The final answer is $\boxed{5}$. | 5 | \boxed{5} | suspected | the source answer is contradicted at b=1, c=1, a=1/25: the expression is exactly 5*sqrt(51)*(sqrt(101) + 20)/51, the source answer gives 5 - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wrong printed answer | 14 | ||
dgn.core.1.084 | 84 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{\sqrt{(2p+1)^3}+\sqrt{(2p-1)^3}}{\sqrt{4p+2\sqrt{4p^2-1}}}.$$ | $4p-\sqrt{4p^2-1}$.
The final answer is $\boxed{4p-\sqrt{4p^2-1}}$. | 4p-\sqrt{4p^2-1} | \boxed{4p-\sqrt{4p^2-1}} | verified | exact (root-free substitution) | 14 | ||
dgn.core.1.085 | 85 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$1-\frac{\frac{1}{\sqrt{a-1}}-\sqrt{a+1}}{\frac{1}{\sqrt{a+1}}-\frac{1}{\sqrt{a-1}}}:\frac{\sqrt{a+1}\sqrt{a^2-1}}{(a-1)\sqrt{a+1}-(a+1)\sqrt{a-1}}.$$ | $\sqrt{a^2-1}$.
The final answer is $\boxed{\sqrt{a^2-1}}$. | \sqrt{a^2-1} | \boxed{\sqrt{a^2-1}} | verified | exact (root-free substitution) | 14 | ||
dgn.core.1.086 | 86 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(\frac{a+2}{\sqrt{2a}}-\frac{a}{\sqrt{2a+2}}+\frac{2}{a-\sqrt{2a}}\right)\frac{\sqrt{a}-\sqrt{2}}{a+2}.$$ | $\frac{1}{\sqrt{a}+\sqrt{2}}$.
The final answer is $\boxed{\frac{1}{\sqrt{a}+\sqrt{2}}}$. | \frac{1}{\sqrt{a}+\sqrt{2}} | \boxed{\frac{1}{\sqrt{a}+\sqrt{2}}} | suspected | the source answer is contradicted at a=1: the expression is exactly -1/2 + 2*sqrt(2)/3, the source answer gives -1 + sqrt(2) - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wrong printed answer | 14 | ||
dgn.core.1.087 | 87 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(\sqrt[4]{36mn^2p} + m\sqrt[3]{\dfrac{3n}{m}} + \sqrt[3]{3np}\right)\left(\sqrt[4]{36mn^2p} - \sqrt[3]{3mn - p}\sqrt{\dfrac{3n}{p}}\right).$$ | $-3n(m+p)$.
The final answer is $\boxed{-3n(m+p)}$. | -3n(m+p) | \boxed{-3n(m+p)} | suspected | the source answer is contradicted at m=1, n=1, p=1: the expression is exactly (sqrt(6) + 2*3**(1/3))*(-2**(1/3)*sqrt(3) + sqrt(6)), the source answer gives -6 - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wrong printed answer | 14 | ||
dgn.core.1.089 | 89 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(\frac{\sqrt{a}}{2}-\frac{1}{2\sqrt{a}}\right)^2\left(\frac{\sqrt{a}-1}{\sqrt{a+1}}-\frac{\sqrt{a+1}}{\sqrt{a-1}}\right).$$ | $(1-a)/\sqrt{a}$.
The final answer is $\boxed{(1-a)/\sqrt{a}}$. | (1-a)/\sqrt{a} | \boxed{(1-a)/\sqrt{a}} | suspected | the source answer is contradicted at a=2: the expression is exactly sqrt(3)*(-4 + sqrt(2))/24, the source answer gives -sqrt(2)/2 - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wrong printed answer | 14 | ||
dgn.core.1.090 | 90 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{1}{2(1+\sqrt{a})}+\frac{1}{2(1-\sqrt{a})}-\frac{a^2+2}{1-a^3}.$$ | $-1/(a^2+a+1)$.
The final answer is $\boxed{-1/(a^2+a+1)}$. | -1/(a^2+a+1) | \boxed{-1/(a^2+a+1)} | verified | numeric(8 pts) | 14 | ||
dgn.core.1.091 | 91 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{\sqrt{2}(x-a)}{2x-a}-\left(\left(\frac{\sqrt{x}}{\sqrt{2x}+\sqrt{a}}\right)^2+\left(\frac{\sqrt{2x}+\sqrt{a}}{2\sqrt{a}}\right)^{-1/2}\right);\quad a=0.32,\ x=0.08.$$ | $1$.
The final answer is $\boxed{1}$. | 1 | \boxed{1} | disputed | third read of the page disagrees with the bulk transcription (still_unresolved) | 14 | ||
dgn.core.1.092 | 92 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(\frac{^{\,4}\!\sqrt{x^3}-^{\,4}\!\sqrt{x}}{1-\sqrt{x}}+\frac{1+\sqrt{x}}{^{\,4}\!\sqrt{x}}\right)^2+\left(1+\frac{2}{\sqrt{x}}+\frac{1}{x}\right)^{-1/2}.$$ | $\frac{1-\sqrt{x}}{1-x}$.
The final answer is $\boxed{\frac{1-\sqrt{x}}{1-x}}$. | \frac{1-\sqrt{x}}{1-x} | \boxed{\frac{1-\sqrt{x}}{1-x}} | unverifiable | unparsed question | 14 | ||
dgn.core.1.093 | 93 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{\sqrt{1-x^2}-1}{x}\left(\frac{1-x}{\sqrt{1-x^2}+x-1}+\frac{\sqrt{1+x}}{\sqrt{1+x}-\sqrt{1-x}}\right).$$ | $-1$.
The final answer is $\boxed{-1}$. | -1 | \boxed{-1} | verified | numeric(8 pts) | 14 | ||
dgn.core.1.094 | 94 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(\frac{pq^3}{(p+q)^{5/2}}-\frac{2pq^2}{(p+q)^{3/2}}+\frac{pq}{\sqrt{p+q}}\right):\left(\frac{p^2}{(p+q)^{5/2}}-\frac{p^2q}{(p+q)^{7/2}}\right).$$ | $q(p+q)$.
The final answer is $\boxed{q(p+q)}$. | q(p+q) | \boxed{q(p+q)} | verified | numeric(8 pts) | 14 | ||
dgn.core.1.095 | 95 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Calculate: $$\frac{(a^2-b^2)(a^2+\sqrt[3]{b^2}+a\sqrt[3]{b})}{a\sqrt[3]{b}+a\sqrt{a-b^3}-\sqrt{b}-\sqrt{ab^2}}:\frac{a^3-b}{a\sqrt[3]{b}-\sqrt[6]{a^3b^2}-\sqrt[3]{b^2}+a\sqrt{a}},$$ if $$a=4.91,\quad b=0.09.$$ | $5$.
The final answer is $\boxed{5}$. | 5 | \boxed{5} | unverifiable | multi-block | 15 | ||
dgn.core.1.096 | 96 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left((1-x^2)^{-1/2}+1+\frac{1}{(1-x^2)^{-1/2}-1}\right)^{-2}:\left(2-x^2-2\sqrt{1-x^2}\right).$$ | $1-x^2$.
The final answer is $\boxed{1-x^2}$. | 1-x^2 | \boxed{1-x^2} | verified | numeric(8 pts) | 15 | ||
dgn.core.1.098 | 98 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(\frac{\sqrt[3]{x}+y}{\sqrt[3]{x-y}}+\frac{\sqrt[3]{x-y}}{\sqrt[3]{x}+y}-2\right):\left(\frac{1}{\sqrt[3]{x-y}}-\frac{1}{\sqrt[3]{x}+y}\right).$$ | $\sqrt[3]{x+y}-\sqrt[3]{x-y}$.
The final answer is $\boxed{\sqrt[3]{x+y}-\sqrt[3]{x-y}}$. | \sqrt[3]{x+y}-\sqrt[3]{x-y} | \boxed{\sqrt[3]{x+y}-\sqrt[3]{x-y}} | suspected | the source answer is contradicted at x=2, y=1: the expression is exactly 2**(1/3), the source answer gives -1 + 3**(1/3) - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wrong printed answer | 15 | ||
dgn.core.1.099 | 99 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(\frac{\frac{4}{a+\frac{1}{b+\frac{1}{c}}}}:\frac{1}{a+\frac{1}{b}}-\frac{4}{b(abc+a+c)}\right)^{-1/2}.$$ | $1/2$.
The final answer is $\boxed{1/2}$. | 1/2 | \boxed{1/2} | unverifiable | unparsed question | 15 | ||
dgn.core.1.100 | 100 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{\sqrt[4]{x^5}+\sqrt[4]{xy^4}-\sqrt[4]{x^4y}-\sqrt[4]{y^5}}{\sqrt{x}-\sqrt{y}}(\sqrt[4]{x}+\sqrt[4]{y}).$$ | $x+y$.
The final answer is $\boxed{x+y}$. | x+y | \boxed{x+y} | verified | exact (root-free substitution) | 15 | ||
dgn.core.1.101 | 101 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{(\sqrt{x}+2)\left(\frac{2}{\sqrt{x}}-1\right)-(\sqrt{x}-2)\left(\frac{2}{\sqrt{x}}+1\right)-\frac{8}{\sqrt{x}}}{(2-\sqrt{x}+2):\left(\sqrt{\frac{2}{x}}+1-\frac{2}{\sqrt{x}}\right)}.$$ | $2$.
The final answer is $\boxed{2}$. | 2 | \boxed{2} | suspected | the source answer is contradicted at x=1: the expression is exactly 2/3 - 2*sqrt(2)/3, the source answer gives 2 - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wrong printed answer | 15 | ||
dgn.core.1.102 | 102 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{1-\sqrt{2t}}{\frac{1-\sqrt[4]{8t^3}}{1-\sqrt[4]{2t}}-\sqrt{2t}}\left(\frac{\sqrt[4]{\frac{1}{2t}}+\sqrt[4]{4t^2}}{1+\sqrt[4]{\frac{1}{2t}}}-\sqrt{2t}\right)^{-1}.$$ | $1$.
The final answer is $\boxed{1}$. | 1 | \boxed{1} | verified | numeric(8 pts) | 15 | ||
dgn.core.1.103 | 103 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{(x^{2/3}+2\sqrt[3]{xy}+4y^{2/3})}{(\sqrt[3]{x^4}-8y\sqrt[3]{x}):\sqrt[3]{xy}}\left(2-\sqrt[3]{\frac{x}{y}}\right).$$ | $-1$.
The final answer is $\boxed{-1}$. | -1 | \boxed{-1} | verified | numeric(5 pts) | 15 | ||
dgn.core.1.104 | 104 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{(z-z\sqrt{z}+2-2\sqrt{z})^2(1+\sqrt{z})^2}{z-2+\frac{1}{z}}-z\sqrt{z}\sqrt{\frac{4}{z}+4+z}.$$ | $z(z+1)(z+2)$.
The final answer is $\boxed{z(z+1)(z+2)}$. | z(z+1)(z+2) | \boxed{z(z+1)(z+2)} | verified | numeric(8 pts) | 15 | ||
dgn.core.1.105 | 105 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(\frac{1}{a+\sqrt{2}}-\frac{a^2+4}{a^3+2\sqrt{2}}\right):\left(\frac{a}{2}-\frac{1}{\sqrt{2}}+\frac{1}{a}\right)^{-1}.$$ | $-\sqrt{2}/(2a)$.
The final answer is $\boxed{-\sqrt{2}/(2a)}$. | -\sqrt{2}/(2a) | \boxed{-\sqrt{2}/(2a)} | verified | exact (simplify) | 15 | ||
dgn.core.1.107 | 107 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(\sqrt{ab}-ab\left(a+\sqrt{ab}\right)^{-1}\right):\left(2\left((ab)^{1/p}-b\right)(a-b)^{-1}\right).$$ | $a/2$.
The final answer is $\boxed{a/2}$. | a/2 | \boxed{a/2} | suspected | the source answer is contradicted at a=1, b=2, p=3: the expression is exactly sqrt(2)/(-2*2**(5/6) - 2*2**(1/3) + 4 + 4*sqrt(2)), the source answer gives 1/2 - reported as suspected, not as a refutation: the difference can come from the reading of the expression, not from a wrong printed answer | 16 | ||
dgn.core.1.110 | 110 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{4a^2-b^2}{a^6-8b^6}\sqrt{a^2-2b\sqrt{a^2-b^2}}\frac{a^4+2a^2b^2+4b^4}{4a^2+4ab+b^2}\sqrt{a^2+2b\sqrt{a^2-b^2}},$$ where $$a=4/3,\quad b=0.25.$$ | $29/35$.
The final answer is $\boxed{29/35}$. | 29/35 | \boxed{29/35} | verified | numeric(8 pts) | 16 | ||
dgn.core.1.112 | 112 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{\sqrt{c-d}}{c^2\sqrt{2c}}\left(\sqrt{\frac{c-d}{c+d}}+\sqrt{\frac{c^2+cd}{c^2-cd}}\right),$$ where $$c=2,\quad d=1/4.$$ | $1/3$.
The final answer is $\boxed{1/3}$. | 1/3 | \boxed{1/3} | verified | numeric(8 pts) | 16 | ||
dgn.core.1.113 | 113 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\frac{x^{3/p}-x^{3/q}}{(x^{1/p}+x^{1/q})^2-2x^{1/q}(x^{1/q}+x^{1/p})}+\frac{x^{1/p}}{x^{(q-p)/pq}+1}.$$ | $\sqrt[p]{x}+\sqrt[q]{x}$.
The final answer is $\boxed{\sqrt[p]{x}+\sqrt[q]{x}}$. | \sqrt[p]{x}+\sqrt[q]{x} | \boxed{\sqrt[p]{x}+\sqrt[q]{x}} | verified | numeric(8 pts) | 16 | ||
dgn.core.1.114 | 114 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$\left(\frac{9-4a^{-2}}{3a^{-1/2}+2a^{-3/2}}-\frac{1+a^{-1}-6a^{-2}}{a^{-1/2}+3a^{-3/2}}\right)^4.$$ | $16a^2$.
The final answer is $\boxed{16a^2}$. | 16a^2 | \boxed{16a^2} | verified | numeric(8 pts) | 16 | ||
dgn.core.1.115 | 115 | core | dgn_math | DGN Math Benchmark | 1 | Identical transformations of algebraic expressions | Simplify the expression: $$4ab+\frac{\left(1+\left(\frac{a}{b}\right)^{-3}\right)a^3}{(\sqrt{a}+\sqrt{b})^2-2\sqrt{ab}}-\frac{\left(\frac{\sqrt{a}+\sqrt{b}}{2b\sqrt{a}}\right)^{-1}+\left(\frac{\sqrt{a}+\sqrt{b}}{2a\sqrt{b}}\right)^{-1}}{\left(\frac{a+\sqrt{ab}}{2}\right)^{-1}+\left(\frac{b+\sqrt{ab}}{2}\right)^{-1}}.$$ | $(a+b)^2$.
The final answer is $\boxed{(a+b)^2}$. | (a+b)^2 | \boxed{(a+b)^2} | verified | numeric(5 pts) | 16 |
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