Questions: Divide the rational expressions. (2r+8)/(r+2) ÷ (4r^2-64) =

Solution Steps

To divide the rational expressions, we first rewrite the division as multiplication by the reciprocal. Then, we factor both the numerator and the denominator of the resulting expression. Finally, we simplify the expression by canceling out common factors.

To divide the rational expressions, we first rewrite the division as multiplication by the reciprocal. The original expression is:

\[ \frac{2r + 8}{r + 2} \div (4r^2 - 64) \]

This can be rewritten as:

\[ \frac{2r + 8}{r + 2} \times \frac{1}{4r^2 - 64} \]

Next, we factor the expressions in the numerator and the denominator:

• The numerator \(2r + 8\) can be factored as \(2(r + 4)\).
• The denominator \(r + 2\) is already in its simplest form.
• The divisor \(4r^2 - 64\) can be factored as \(4(r - 4)(r + 4)\).

Now, substitute the factored forms back into the expression:

\[ \frac{2(r + 4)}{r + 2} \times \frac{1}{4(r - 4)(r + 4)} \]

Cancel the common factor \((r + 4)\) from the numerator and the divisor:

\[ \frac{2}{r + 2} \times \frac{1}{4(r - 4)} \]

This simplifies to:

\[ \frac{1}{2(r - 4)(r + 2)} \]

Final Answer