Question

divide.
\\(\frac{x + 4}{2x + 16} \div \frac{5x + 20}{8x + 64}\\)
simplify your answer as much as possible.

Explanation:

Step1: Rewrite division as multiplication

To divide two fractions, we multiply the first fraction by the reciprocal of the second fraction. So, \(\frac{x + 4}{2x + 16}\div\frac{5x + 20}{8x + 64}=\frac{x + 4}{2x + 16}\times\frac{8x + 64}{5x + 20}\)

Step2: Factor each expression

Factor the numerators and denominators:

• For \(2x + 16\), we can factor out a 2: \(2x+16 = 2(x + 8)\)
• For \(8x + 64\), we can factor out an 8: \(8x+64=8(x + 8)\)
• For \(5x + 20\), we can factor out a 5: \(5x + 20=5(x + 4)\)

Substituting these factored forms back into the expression, we get: \(\frac{x + 4}{2(x + 8)}\times\frac{8(x + 8)}{5(x + 4)}\)

Step3: Cancel out common factors

We can cancel out the common factors \((x + 4)\) and \((x + 8)\) from the numerator and the denominator:
\(\frac{\cancel{x + 4}}{2\cancel{(x + 8)}}\times\frac{8\cancel{(x + 8)}}{5\cancel{(x + 4)}}\)
After canceling, we are left with \(\frac{8}{2\times5}\)

Step4: Simplify the remaining fraction

Simplify \(\frac{8}{10}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2. So, \(\frac{8\div2}{10\div2}=\frac{4}{5}\)

Answer:

\(\frac{4}{5}\)