Sure! Let's go through the process of simplifying each expression step-by-step.
### Simplifying [tex]\(4^{-\frac{3}{2}}\)[/tex]
1. Understanding the exponent: The exponent [tex]\(-\frac{3}{2}\)[/tex] can be broken into the product of two separate exponents.
[tex]\[
4^{-\frac{3}{2}} = \left(4^{\frac{1}{2}}\right)^{-3}
\][/tex]
2. Calculate the first step: [tex]\(4^{\frac{1}{2}}\)[/tex] is equivalent to the square root of 4.
[tex]\[
4^{\frac{1}{2}} = \sqrt{4} = 2
\][/tex]
3. Apply the negative exponent: Now raise 2 to the power of [tex]\(-3\)[/tex].
[tex]\[
\left(2\right)^{-3} = \frac{1}{2^3} = \frac{1}{8}
\][/tex]
Thus, we have:
[tex]\[
4^{-\frac{3}{2}} = 0.125
\][/tex]
### Simplifying [tex]\(\left(\frac{1}{16}\right)^{-\frac{5}{4}}\)[/tex]
1. Understanding the base and exponent: The negative exponent indicates a reciprocal. So:
[tex]\[
\left(\frac{1}{16}\right)^{-\frac{5}{4}} = 16^{\frac{5}{4}}
\][/tex]
2. Breaking down the exponent: The exponent [tex]\(\frac{5}{4}\)[/tex] can be written as [tex]\(1.25\)[/tex], which is also [tex]\(\left(16^{\frac{1}{4}}\right)^5\)[/tex].
3. Calculate intermediate steps: First, find [tex]\(16^{\frac{1}{4}}\)[/tex].
[tex]\[
16^{\frac{1}{4}} = \sqrt[4]{16} = 2
\][/tex]
Since [tex]\(2^4 = 16\)[/tex].
4. Raise the intermediate result to the power of 5:
[tex]\[
2^5 = 32
\][/tex]
Thus, we have:
[tex]\[
\left(\frac{1}{16}\right)^{-\frac{5}{4}} = 32.0
\][/tex]
So, the simplified forms of the given expressions are:
[tex]\[
4^{-\frac{3}{2}} = 0.125
\][/tex]
[tex]\[
\left(\frac{1}{16}\right)^{-\frac{5}{4}} = 32.0
\][/tex]
