Certainly! Let's evaluate the given expression step by step:
[tex]\[
\left(25^{-\frac{3}{2}}\right)^{\frac{1}{3}}
\][/tex]
### Step 1: Evaluate the inner exponentiation [tex]\(25^{-\frac{3}{2}}\)[/tex]
First, we need to simplify [tex]\(25^{-\frac{3}{2}}\)[/tex].
- The base is 25.
- The exponent is [tex]\(-\frac{3}{2}\)[/tex].
Let's break down the exponent:
[tex]\[ -\frac{3}{2} = -1 \cdot \frac{3}{2} \][/tex]
[tex]\[ = -1 \cdot \left(\frac{3}{2}\right) \][/tex]
We can interpret [tex]\(25^{-\frac{3}{2}}\)[/tex] as:
[tex]\[ 25^{-\frac{3}{2}} = \frac{1}{25^{\frac{3}{2}}} \][/tex]
Next, let's compute [tex]\(25^{\frac{3}{2}}\)[/tex]:
- [tex]\(25\)[/tex] can be written as [tex]\(5^2\)[/tex].
[tex]\[ 25^{\frac{3}{2}} = (5^2)^{\frac{3}{2}} \][/tex]
Now use the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[ (5^2)^{\frac{3}{2}} = 5^{2 \cdot \frac{3}{2}} = 5^3 = 125 \][/tex]
Hence,
[tex]\[ 25^{-\frac{3}{2}} = \frac{1}{125} = 0.008 \][/tex]
### Step 2: Evaluate the outer exponentiation [tex]\(\left(25^{-\frac{3}{2}}\right)^{\frac{1}{3}}\)[/tex]
Now that we have [tex]\(25^{-\frac{3}{2}} = 0.008\)[/tex], we need to compute:
[tex]\[ \left(0.008\right)^{\frac{1}{3}} \][/tex]
Taking the cube root of 0.008 (since [tex]\(\frac{1}{3}\)[/tex] represents the cube root):
[tex]\[ \left(0.008\right)^{\frac{1}{3}} = 0.2 \][/tex]
### Conclusion
Thus, the value of the given expression is:
[tex]\[ \left(25^{-\frac{3}{2}}\right)^{\frac{1}{3}} = 0.2 \][/tex]
So the final answer is:
[tex]\[ \boxed{0.2} \][/tex]
