Question: Divide $\frac{32r + 4}{r} \div \frac{8r + 1}{r^3}$. $\frac{32r + 4}{r} \div \frac{8r +…

Divide $\frac{32r + 4}{r} \div \frac{8r + 1}{r^3}$.

$\frac{32r + 4}{r} \div \frac{8r + 1}{r^3} = \boxed{}$

(Type your answer using exponential notation. Simplify your answer.)

Solution

Step 1: Rewrite the division as multiplication by the reciprocal. \[ \frac{32r + 4}{r} \div \frac{8r + 1}{r^3} = \frac{32r + 4}{r} \times \frac{r^3}{8r + 1} \] Step 2: Factor the numerator of the first fraction. \[ 32r + 4 = 4(8r + 1) \] Step 3: Substitute the factored form back into the equation. \[ \frac{4(8r + 1)}{r} \times \frac{r^3}{8r + 1} \] Step 4: Cancel the common term \((8r + 1)\). \[ 4 \times \frac{r^3}{r} \] Step 5: Simplify the exponents. \[ 4r^{2} \] The simplified answer is \(4r^{2}\).