- Simplify the numerator: \((2 a^4)^2 = 4a^8\).
- Simplify the denominator: \(a^0 a^5 = a^5\).
- Simplify the fraction: \(\frac{4a^8}{a^5} = 4a^3\).
- Substitute \(a = -5\): \(4(-5)^3 = 4(-125) = \boxed{-500}\).
Explanation
- Simplifying the Expression
We are given the expression \(\frac{(2 a^4)^2}{a^0 a^5}\) and asked to simplify it using the laws of exponents and then evaluate it at \(a = -5\).
First, let's analyze the components of the expression:
- The numerator is \((2 a^4)^2\). We apply the power of a product rule \((xy)^n = x^n y^n\) and the power of a power rule \((x^m)^n = x^{mn}\).
\[(2 a^4)^2 = 2^2 \cdot (a^4)^2 = 4 \cdot a^{4 \times 2} = 4a^8\]
- The denominator is \(a^0 a^5\). We apply the zero exponent rule \(x^0 = 1\) (where \(a
eq 0\)).
\[a^0 a^5 = 1 \cdot a^5 = a^5\]
Putting these together, the original expression simplifies to:
\[\frac{4a^8}{a^5}\]
- Applying the Quotient Rule
Now we further simplify \(\frac{4a^8}{a^5}\) using the quotient rule for exponents \(\frac{x^m}{x^n} = x^{m-n}\):
\[\frac{4a^8}{a^5} = 4a^{8-5} = 4a^3\]
This is our simplified expression.
- Evaluating the Expression
Finally, we evaluate the expression \(4a^3\) at \(a = -5\):
\[4(-5)^3 = 4 \cdot (-5 \cdot -5 \cdot -5) = 4 \cdot (-125) = -500\]
Comparing this result to the given options:
The correct value is \(-500\).
Examples
The rules of exponents are crucial in scientific notation and engineering when working with very large or very small numbers. For example, if you are calculating the volume of a very small object in nanometers or the distance between stars in kilometers, these properties allow you to simplify complex calculations into manageable forms. Understanding how to manipulate powers ensures accuracy in modeling physical systems efficiently.