Chapter A.10, Problem 106AYU

To determine

Factor the expression $2x{\left(3x+4\right)}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+x^2.4{\left(3x+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$ and express the answer so that only positive exponents occur.

Answer to Problem 106AYU

The factorized form of the expression $2x{\left(3x+4\right)}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+x^2.4{\left(3x+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$ is $2x{\left(3x+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(5x+4\right)$.

Explanation of Solution

Given:

It is asked to determine the factor of each expression and express the answer that contains only positive exponents.

Calculation:

Consider $2x{\left(3x+4\right)}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+x^2.4{\left(3x+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$,

$2x{\left(3x+4\right)}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+x^2.4{\left(3x+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

$=2x{\left(3x+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(3x+4+2x\right)$

$=2x{\left(3x+4\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\left(5x+4\right)$