Question

1. multiply the rational expressions. write your answer in its fully factored form. \\( \left( \frac{6b^2 - 3b}{24b - 6} \

ight) \left( \frac{6b - 48}{6b^2 + 9b} \
ight) \\) show your work here hint: to add an exponent (\\( x^n \\)), type \exponent\ or press \^\ \\( \frac{(2b - 1)(b - 8)}{(4b - 1)(4b + 3)} \\) \\( \frac{(2b - 1)(b - 64)}{(4b - 1)(2b + 3)} \\) \\( \frac{(2b - 1)(b + 8)}{(4b - 1)(4b + 3)} \\) \\( \frac{(2b - 1)(b - 8)}{(4b - 1)(2b + 3)} \\)

Explanation:

Step1: Factor each numerator and denominator

Factor \(6b^2 - 3b\): \(3b(2b - 1)\)
Factor \(24b - 6\): \(6(4b - 1)\)
Factor \(6b - 48\): \(6(b - 8)\)
Factor \(6b^2 + 9b\): \(3b(2b + 3)\)

Step2: Substitute factored forms into the expression

The expression becomes \(\frac{3b(2b - 1)}{6(4b - 1)} \cdot \frac{6(b - 8)}{3b(2b + 3)}\)

Step3: Cancel common factors

Cancel \(3b\) from numerator and denominator, cancel \(6\) from numerator and denominator:
\(\frac{(2b - 1)}{(4b - 1)} \cdot \frac{(b - 8)}{(2b + 3)}\)

Step4: Multiply the remaining factors

Multiply the numerators and denominators: \(\frac{(2b - 1)(b - 8)}{(4b - 1)(2b + 3)}\)

Answer:

\(\frac{(2b - 1)(b - 8)}{(4b - 1)(2b + 3)}\) (corresponding to the last option: \(\boldsymbol{\frac{(2b - 1)(b - 8)}{(4b - 1)(2b + 3)}}\))