To evaluate the expression [tex]\(\left(25^{-\frac{3}{2}}\right)^{\frac{1}{3}}\)[/tex], let's break it down into manageable steps:
1. Exponent Simplification:
- First, we need to compute the inside expression [tex]\( 25^{-\frac{3}{2}} \)[/tex].
2. Computing [tex]\( 25^{-\frac{3}{2}} \)[/tex]:
- Recall that [tex]\( 25 \)[/tex] can be written as [tex]\( 5^2 \)[/tex].
- Therefore, [tex]\( 25^{-\frac{3}{2}} \)[/tex] can be rewritten using the properties of exponents as [tex]\( (5^2)^{-\frac{3}{2}} \)[/tex].
- Using the rule of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we get [tex]\( (5^2)^{-\frac{3}{2}} = 5^{2 \cdot -\frac{3}{2}} = 5^{-3} \)[/tex].
- This further simplifies to [tex]\( 1 / 5^3 = 1 / 125 \)[/tex].
3. Intermediate Result:
- The evaluation of [tex]\( 25^{-\frac{3}{2}} \)[/tex] gives us [tex]\( 0.008 \)[/tex] (since [tex]\( 1 / 125 = 0.008 \)[/tex]).
4. Next Exponentiation:
- Now, we need to calculate the expression [tex]\((0.008)^{\frac{1}{3}}\)[/tex], which is the cube root of [tex]\( 0.008 \)[/tex].
5. Computing Cube Root:
- The cube root of [tex]\( 0.008 \)[/tex] can be found either by recognizing it as [tex]\( (2/10)^3 = (1/5)^3 \)[/tex] or directly evaluating [tex]\( 0.008^{\frac{1}{3}} \)[/tex].
- The result of this calculation is [tex]\( 0.2 \)[/tex].
6. Final Result:
- Therefore, [tex]\(\left(25^{-\frac{3}{2}}\right)^{\frac{1}{3}}\)[/tex] evaluates to [tex]\( 0.2 \)[/tex].
So, the final answer is:
[tex]\[
\left(25^{-\frac{3}{2}}\right)^{\frac{1}{3}} = 0.2
\][/tex]
