Perform the indicated operation involving rational expressions. $\frac { 8 a ^ { 2 } - 64 a } { a ^ { 3 } - 8 a ^ { 2 } } \cdot \frac { a ^ { 2 } - 16 a + 64 } { a ^ { 2 } - 64 }$
A) $\frac { 8 a ^ { 2 } + 64 } { a ^ { 2 } + 8 a }$
B) $\frac { a - 8 } { a ^ { 2 } + 8 a }$
C) $\frac { 8 } { a ^ { 2 } + 8 a }$
D) $\frac { 8 a - 64 } { a ^ { 2 } + 8 a }$

Question

Quizplus · Accepted Answer

First, we factor both the numerator and denominator of the first fraction:

$$\frac{8a^2-64a}{a^3-8a^2} = \frac{8a(a-8)}{a^2(a-8)} = \frac{8(a-8)}{a}$$

Then, we factor both the numerator and denominator of the second fraction:

$$\frac{a^2-16a+64}{a^2-64} = \frac{(a-8)(a-8)}{(a-8)(a+8)} = \frac{a-8}{a+8}$$

Multiplying the two fractions, we get:

$$\frac{8(a-8)}{a} \cdot \frac{a-8}{a+8} = \frac{8(a-8)(a-8)}{a(a+8)} = \frac{8(a^2-16a+64)}{a(a+8)}$$

We can simplify this expression further by factoring the numerator:

$$\frac{8(a-8)(a-8)}{a(a+8)} = \frac{8(a-8)^2}{a(a+8)} = \frac{8(a-8)(a-8)}{a(a+8)}$$

This expression cannot be simplified any further, so the correct answer is D: $\frac{8a-64}{a^2+8a}$.

Perform the Indicated Operation Involving Rational Expressions 8a2−64aa3−8a2⋅a2−16a+64a2−64\frac { 8 a ^ { 2 } - 64 a } { a ^ { 3 } - 8 a ^ { 2 } } \cdot \frac { a ^ { 2 } - 16 a + 64 } { a ^ { 2 } - 64 }a3−8a28a2−64a​⋅a2−64a2−16a+64​

Perform the Indicated Operation Involving Rational Expressions $\frac { 8 a ^ { 2 } - 64 a } { a ^ { 3 } - 8 a ^ { 2 } } \cdot \frac { a ^ { 2 } - 16 a + 64 } { a ^ { 2 } - 64 }$