Sure! Let's break down the problem step-by-step.
Given:
[tex]\[ x = 25 \][/tex]
[tex]\[ y = -2 \][/tex]
We need to find the value of:
[tex]\[ \frac{\sqrt[y]{x^y} \cdot x^{3 / 2}}{\sqrt{x}} \][/tex]
First, calculate [tex]\( x^y \)[/tex]:
[tex]\[ x^y = 25^{-2} \][/tex]
[tex]\[ 25^{-2} = \frac{1}{25^2} = \frac{1}{625} = 0.0016 \][/tex]
Next, find [tex]\( \sqrt[y]{x^y} \)[/tex] which is the y-th root of [tex]\( x^y \)[/tex]:
[tex]\[ \sqrt[y]{x^y} = (x^y)^{1/y} \][/tex]
Since [tex]\( y = -2 \)[/tex],
[tex]\[ (x^y)^{1/y} = (0.0016)^{1/-2} = 25 \][/tex]
Now, calculate [tex]\( x^{3/2} \)[/tex]:
[tex]\[ x^{3/2} = 25^{3/2} \][/tex]
[tex]\[ 25 = 5^2 \rightarrow 25^{3/2} = (5^2)^{3/2} = 5^3 = 125 \][/tex]
Next, find [tex]\( \sqrt{x} \)[/tex]:
[tex]\[ \sqrt{x} = x^{1/2} \][/tex]
[tex]\[ 25^{1/2} = \sqrt{25} = 5 \][/tex]
Now we combine all parts into the expression [tex]\( \frac{\sqrt[y]{x^y} \cdot x^{3/2}}{\sqrt{x}} \)[/tex]:
[tex]\[ \frac{\sqrt[y]{x^y} \cdot x^{3/2}}{\sqrt{x}} = \frac{25 \cdot 125}{5} \][/tex]
Finally, calculate the result:
[tex]\[ 25 \cdot 125 = 3125 \][/tex]
[tex]\[ \frac{3125}{5} = 625 \][/tex]
So, the value of the given expression is:
[tex]\[
\boxed{625}
\][/tex]
