Sure, let's solve the expression [tex]\((\sqrt{16})^{-\frac{3}{2}}\)[/tex] step-by-step.
1. Evaluate the inner square root:
[tex]\[
\sqrt{16}
\][/tex]
The square root of 16 is 4.
2. Substitute the result of the square root back into the original expression:
[tex]\[
(4)^{-\frac{3}{2}}
\][/tex]
3. Apply the exponentiation. To do this, let's break down the exponent [tex]\(-\frac{3}{2}\)[/tex] into parts:
- The negative sign indicates that we take the reciprocal of the base.
- The exponent [tex]\(\frac{3}{2}\)[/tex] means taking the square root and then cubing the result (or vice versa).
So, we need to evaluate [tex]\( (4)^{-\frac{3}{2}} \)[/tex]:
- First, find the square root of 4:
[tex]\[
\sqrt{4} = 2
\][/tex]
- Then, raise 2 to the power of 3:
[tex]\[
2^3 = 8
\][/tex]
- Since we have a negative exponent, take the reciprocal of this result:
[tex]\[
8^{-1} = \frac{1}{8}
\][/tex]
Thus, [tex]\((\sqrt{16})^{-\frac{3}{2}} = \frac{1}{8} = 0.125\)[/tex].
So, the detailed solution of [tex]\((\sqrt{16})^{-\frac{3}{2}}\)[/tex] yields 0.125.
