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
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
