Let's solve the given expression step-by-step using the properties of exponents.
We start with the expression:
[tex]\[
\frac{\left(2 a^4\right)^2}{a^0 a^5}
\][/tex]
### Step 1: Simplify the Numerator
First, simplify the numerator [tex]\(\left(2 a^4\right)^2\)[/tex]:
[tex]\[
\left(2 a^4\right)^2 = 2^2 \cdot \left(a^4\right)^2
\][/tex]
Using the properties of exponents, we know that [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
2^2 = 4
\][/tex]
[tex]\[
\left(a^4\right)^2 = a^{4 \cdot 2} = a^8
\][/tex]
So, the numerator simplifies to:
[tex]\[
4 a^8
\][/tex]
### Step 2: Simplify the Denominator
Now, simplify the denominator [tex]\(a^0 a^5\)[/tex]:
Using the property that any number to the power of zero is 1:
[tex]\[
a^0 = 1
\][/tex]
So, the denominator simplifies to:
[tex]\[
1 \cdot a^5 = a^5
\][/tex]
### Step 3: Combine the Simplified Numerator and Denominator
We now have:
[tex]\[
\frac{4 a^8}{a^5}
\][/tex]
Using the properties of exponents (namely [tex]\(a^m / a^n = a^{m-n}\)[/tex]):
[tex]\[
\frac{4 a^8}{a^5} = 4 a^{8-5} = 4 a^3
\][/tex]
### Step 4: Substitute [tex]\(a = -5\)[/tex] into the Expression
Now, we substitute [tex]\(a = -5\)[/tex] into the simplified expression [tex]\(4 a^3\)[/tex]:
[tex]\[
4 \left(-5\right)^3
\][/tex]
Calculate [tex]\((-5)^3\)[/tex]:
[tex]\[
(-5)^3 = -125
\][/tex]
Multiply by 4:
[tex]\[
4 \cdot (-125) = -500
\][/tex]
Therefore, the value of the expression when [tex]\(a = -5\)[/tex] is:
[tex]\[
-500
\][/tex]
### Conclusion
The correct answer is:
A. -500
