Sure, let's simplify the given expression step by step.
We start with the expression:
[tex]\[
\left(\frac{-2 y^2}{z}\right)^5
\][/tex]
First, apply the power of 5 to both the numerator and the denominator inside the parentheses. This is a property of exponents:
[tex]\[
\left(\frac{-2 y^2}{z}\right)^5 = \frac{(-2 y^2)^5}{z^5}
\][/tex]
Next, focus on the numerator [tex]\((-2 y^2)^5\)[/tex]. We can distribute the exponent to both [tex]\(-2\)[/tex] and [tex]\(y^2\)[/tex]:
[tex]\[
(-2 y^2)^5 = (-2)^5 \cdot (y^2)^5
\][/tex]
Now, calculate [tex]\((-2)^5\)[/tex]:
[tex]\[
(-2)^5 = -32
\][/tex]
Next, apply the power of 5 to [tex]\(y^2\)[/tex]:
[tex]\[
(y^2)^5 = y^{2 \cdot 5} = y^{10}
\][/tex]
Putting these results together, the numerator becomes:
[tex]\[
(-2)^5 \cdot (y^2)^5 = -32 \cdot y^{10}
\][/tex]
So our expression now is:
[tex]\[
\frac{-32 \cdot y^{10}}{z^5}
\][/tex]
Simplifying this, the final simplified form of the given expression is:
[tex]\[
-32 \cdot \frac{y^{10}}{z^5}
\][/tex]
Or simply:
[tex]\[
-32 \cdot y^{10} / z^5
\][/tex]
So, the simplified expression is:
[tex]\[
-32 \cdot y^{10} / z^5
\][/tex]
