Let's evaluate the given expression step by step.
We start with the expression:
[tex]\[ \left(25^{-\frac{3}{2}}\right)^{\frac{1}{3}} \][/tex]
First, let's deal with the inner part of the expression [tex]\( 25^{-\frac{3}{2}} \)[/tex].
### Step 1: Evaluate [tex]\( 25^{-\frac{3}{2}} \)[/tex]
The base here is 25, and we need to raise it to the power of [tex]\(-\frac{3}{2}\)[/tex].
[tex]\[ 25^{-\frac{3}{2}} \][/tex]
The negative exponent indicates that we take the reciprocal of the base raised to the positive power:
[tex]\[ 25^{-\frac{3}{2}} = \frac{1}{25^{\frac{3}{2}}} \][/tex]
Next, we evaluate [tex]\( 25^{\frac{3}{2}} \)[/tex]. The exponent [tex]\(\frac{3}{2}\)[/tex] can be broken down into two parts: a square root and a cube:
[tex]\[ 25^{\frac{3}{2}} = (25^{\frac{1}{2}})^3 \][/tex]
The square root of 25 is 5:
[tex]\[ 25^{\frac{1}{2}} = \sqrt{25} = 5 \][/tex]
Now we cube the result:
[tex]\[ 5^3 = 125 \][/tex]
So,
[tex]\[ 25^{\frac{3}{2}} = 125 \][/tex]
Then,
[tex]\[ 25^{-\frac{3}{2}} = \frac{1}{125} \][/tex]
We know the value of [tex]\(\frac{1}{125}\)[/tex] is approximately 0.008:
[tex]\[ 25^{-\frac{3}{2}} = 0.008 \][/tex]
### Step 2: Evaluate [tex]\(\left(0.008\right)^{\frac{1}{3}}\)[/tex]
Now we know [tex]\( 25^{-\frac{3}{2}} \)[/tex] is 0.008. Next, we raise this result to the power of [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[ \left(0.008\right)^{\frac{1}{3}} \][/tex]
The cube root of 0.008 is:
[tex]\[ \left(0.008\right)^{\frac{1}{3}} = 0.2 \][/tex]
Thus, the final result of the given expression is:
[tex]\[ \left(25^{-\frac{3}{2}}\right)^{\frac{1}{3}} = 0.2 \][/tex]
