SAT Mathematics Practice Test 3

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