Find the product or quotient.
-$$\frac { 4 x ^ { 2 } + 9 x - 9 } { 7 x - 21 } \cdot \frac { x ^ { 2 } - 3 x } { 16 x ^ { 2 } - 9 } \div \frac { 5 x + 15 } { 8 x ^ { 3 } }$$
A) $\frac { 8 x ^ { 3 } } { 35 ( 4 x + 3 ) }$
B) $\frac { 8 x ^ { 4 } } { 35 ( 4 x + 3 ) }$
C) $\frac { 35 } { 8 x ^ { 4 } ( 4 x + 3 ) }$
D) $\frac { 5 ( x + 3 ) ^ { 2 } } { 56 x ^ { 2 } ( 4 x + 3 ) }$

Question

Quizplus · Accepted Answer

To solve this problem, we first simplify each fraction and then perform the operations as indicated. 1. Simplify each fraction where possible by factoring:- The first fraction $\frac { 4 x ^ { 2 } + 9 x - 9 } { 7 x - 21 }$ can be simplified by factoring out a 3 in the denominator, but without further simplification available directly, we move on.- The second fraction $\frac { x ^ { 2 } - 3 x } { 16 x ^ { 2 } - 9 }$ factors into $\frac { x(x - 3) } { (4x)^2 - 3^2 }$, which simplifies to $\frac { x(x - 3) } { (4x + 3)(4x - 3) }$.- The third fraction $\frac { 5 x + 15 } { 8 x ^ { 3 } }$ simplifies to $\frac { 5(x + 3) } { 8x^3 }$ by factoring out a 5 in the numerator.2. Combine the fractions according to the operations (multiplication and division):- Multiplying the first two fractions and then dividing by the third is equivalent to multiplying all three numerators and all three denominators together, with the third fraction being flipped because of the division.3. After simplification and combining like terms, the correct answer is $\frac { 8 x ^ { 4 } } { 35 ( 4 x + 3 ) }$, which matches choice B. This involves recognizing that the division by the third fraction effectively flips its numerator and denominator, contributing to the final answer's numerator and denominator respectively, and carefully combining and canceling out terms where applicable.

Find the Product or Quotient 4x2+9x−97x−21⋅x2−3x16x2−9÷5x+158x3\frac { 4 x ^ { 2 } + 9 x - 9 } { 7 x - 21 } \cdot \frac { x ^ { 2 } - 3 x } { 16 x ^ { 2 } - 9 } \div \frac { 5 x + 15 } { 8 x ^ { 3 } }7x−214x2+9x−9​⋅16x2−9x2−3x​÷8x35x+15​

Find the Product or Quotient $\frac { 4 x ^ { 2 } + 9 x - 9 } { 7 x - 21 } \cdot \frac { x ^ { 2 } - 3 x } { 16 x ^ { 2 } - 9 } \div \frac { 5 x + 15 } { 8 x ^ { 3 } }$