Select the correct answer.

Use the properties of exponents to rewrite this expression:
[tex]\[
\frac{\left(2 a^4\right)^2}{a^0 a^5}
\][/tex]

What is the value of the rewritten expression when \(a=-5\)?

A. -500
B. -60
C. -250
D. -20



Answer :

To solve the given expression \(\frac{(2 a^4)^2}{a^0 a^5}\) using the properties of exponents, we can follow these steps:

1. Simplify the numerator \((2 a^4)^2\):
- This can be rewritten using the property \((xy)^n = x^n y^n\).
- So, \((2 a^4)^2 = 2^2 \cdot (a^4)^2 = 4 \cdot a^{8}\).

Therefore, the numerator simplifies to \(4 a^8\).

2. Simplify the denominator \(a^0 a^5\):
- Using the property \(a^m \cdot a^n = a^{m+n}\), we have \(a^0 \cdot a^5 = a^{0+5} = a^5\).

Therefore, the denominator simplifies to \(a^5\).

3. Combine the simplified numerator and denominator:
- Now we have \(\frac{4 a^8}{a^5}\).

4. Simplify the fraction using the property \(\frac{a^m}{a^n} = a^{m-n}\):
- Thus, \(\frac{4 a^8}{a^5} = 4 a^{8-5} = 4 a^3\).

5. Substitute \(a = -5\) into the simplified expression:
- We need to find the value of \(4 a^3\) when \(a = -5\).
- So, \(4 (-5)^3 = 4 \cdot (-125) = -500\).

Therefore, the correct answer is:

A. -500