SAT Mathematics Practice Test 3

By definition, ${\mathit{a}}^{\frac{\mathit{m}}{\mathit{n}}}{=}^{\mathit{n}}\sqrt{{\mathit{a}}^{\mathit{m}}}$ for any positive integers $\mathit{m}$ and $\mathit{n}$ .
It follows, therefore ,that ${\mathit{a}}^{\frac{2}{3}}=\sqrt[3]{{\mathit{a}}^2}$
Choice A is incorrect.
By definition ${\mathit{a}}^{\frac{1}{\mathit{n}}}{=}^{\mathit{n}}\sqrt{\mathit{a}}$ for any positive integer n.
Applying this definition as well as the power property of exponents to the expression $\sqrt{{\mathit{a}}^{\frac{1}{3}}}$ yields $\sqrt{{\mathit{a}}^{\frac{1}{3}}}={\left({\mathit{a}}^{\frac{1}{3}}\right)}^{\frac{1}{2}}$.Because ${\mathit{a}}^{\frac{1}{6}}\ne {\mathit{a}}^{\frac{2}{3}},\sqrt{{\mathit{a}}^{\frac{1}{3}}}$ is not the correct answer.
Choice B is incorrect.
By definition,${\mathit{a}}^{\frac{1}{\mathit{n}}}{=}^{\mathit{n}}\sqrt{\mathit{a}}$ for any positive integer n.
Applying this definition as well as the power property of exponents to the expression $\sqrt{{\mathit{a}}^3}$ yields $\sqrt{{\mathit{a}}^3}={\left({\mathit{a}}^3\right)}^{\frac{1}{2}}={\mathit{a}}^{\frac{3}{2}}$.
Because ${\mathit{a}}^{\frac{3}{2}}\ne {\mathit{a}}^{\frac{2}{3}},\sqrt{{\mathit{a}}^3}$ is not the correct answer.
Choice C is incorrect.
By definition, ${\mathit{a}}^{\frac{1}{\mathit{n}}}=\sqrt{\mathit{a}}$ for any positive integer $\mathit{n}$.
Applying this definition as well as the power property of exponents to the expression $\sqrt[3]{{\mathit{a}}^{\frac{1}{2}}}$ yields $\sqrt[3]{{\mathit{a}}^{\frac{1}{2}}}={\left({\mathit{a}}^{\frac{1}{2}}\right)}^{\frac{1}{3}}={\mathit{a}}^{\frac{1}{6}}$ Because ${\mathit{a}}^{\frac{1}{6}}\ne {\mathit{a}}^{\frac{2}{3}},\sqrt[3]{{\mathit{a}}^{\frac{1}{2}}}$ is not the correct answer.