Question

type the correct answer in the box. use numerals instead of words. if necessary, use / for the fraction bar. the quotient of \\(\frac{y^2 - 16}{4y - 16} \div \frac{y + 4}{16y}\\) is .

Explanation:

Step1: Factor numerators and denominators

Factor \(y^2 - 16\) as a difference of squares: \(y^2 - 16=(y + 4)(y - 4)\).
Factor \(4y - 16\) by taking out the common factor 4: \(4y - 16 = 4(y - 4)\).

Step2: Rewrite division as multiplication

Recall that dividing by a fraction is the same as multiplying by its reciprocal. So, \(\frac{y^2 - 16}{4y - 16}\div\frac{y + 4}{16y}=\frac{y^2 - 16}{4y - 16}\times\frac{16y}{y + 4}\).

Step3: Substitute factored forms

Substitute the factored forms into the expression:
\(\frac{(y + 4)(y - 4)}{4(y - 4)}\times\frac{16y}{y + 4}\).

Step4: Cancel common factors

Cancel out the common factors \((y + 4)\) and \((y - 4)\) (assuming \(y
eq - 4\) and \(y
eq4\) to avoid division by zero):
After canceling, we have \(\frac{1}{4}\times\frac{16y}{1}\).

Step5: Multiply the remaining terms

Multiply \(\frac{1}{4}\) and \(16y\): \(\frac{1\times16y}{4}=4y\).

Answer:

\(4y\)