Question

algebra and geometry: negative rational exponents: negative simplify. write your answers without exponents. simplify. write your answers without exponents.
4^{-\frac{5}{2}}=
(\frac{1}{16})^{\frac{3}{4}}=

Explanation:

Step1: Simplify $4^{-\frac{5}{2}}$

Use the rule $a^{-n}=\frac{1}{a^{n}}$ and $a^{\frac{m}{n}}=\sqrt[n]{a^{m}}$. So $4^{-\frac{5}{2}}=\frac{1}{4^{\frac{5}{2}}}$, and $4^{\frac{5}{2}}=\sqrt{4^{5}}=\sqrt{(2^{2})^{5}}=\sqrt{2^{10}} = 2^{5}=32$. Then $\frac{1}{4^{\frac{5}{2}}}=\frac{1}{32}$.

Step2: Simplify $(\frac{1}{16})^{\frac{3}{4}}$

Rewrite $\frac{1}{16}$ as $2^{-4}$. Then $(\frac{1}{16})^{\frac{3}{4}}=(2^{-4})^{\frac{3}{4}}$. Use the power - of - a - power rule $(a^{m})^{n}=a^{mn}$, so $(2^{-4})^{\frac{3}{4}}=2^{-4\times\frac{3}{4}}=2^{-3}=\frac{1}{8}$.

Answer:

$4^{-\frac{5}{2}}=\frac{1}{32}$, $(\frac{1}{16})^{\frac{3}{4}}=\frac{1}{8}$