acomodats89m8

2023-03-30

Perform the indicated operation and simplify the result. Leave your answer in factored form
$\left[\frac{\left(4x-8\right)}{\left(-3x\right)}\right].\left[\frac{12}{\left(12-6x\right)}\right]$

pagarec0pz

To solve the given expression, we'll follow these steps:
Step 1: Simplify the expression inside the brackets.
Step 2: Multiply the two fractions together.
Step 3: Simplify the resulting expression.
Let's begin with step 1:
The expression inside the first bracket is $\frac{4x-8}{-3x}$, and inside the second bracket is $\frac{12}{12-6x}$.
To simplify the first expression, we can factor out a common factor of 4 from the numerator:
$\frac{4\left(x-2\right)}{-3x}$.
For the second expression, there are no common factors to factor out.
Now, let's move on to step 2:
To multiply the two fractions together, we multiply the numerators and the denominators:
$\frac{4\left(x-2\right)}{-3x}·\frac{12}{12-6x}$.
Multiplying the numerators gives us $4\left(x-2\right)·12$, and multiplying the denominators gives us $-3x·\left(12-6x\right)$.
The expression now becomes:
$\frac{4\left(x-2\right)·12}{-3x·\left(12-6x\right)}$.
Moving on to step 3:
Let's simplify the numerator first. Multiplying $4\left(x-2\right)·12$ gives us $48\left(x-2\right)$.
Now, let's simplify the denominator. Multiplying $-3x·\left(12-6x\right)$ can be done by distributing $-3x$ into the parentheses:
$-3x·12+\left(-3x\right)·\left(-6x\right)$.
This simplifies to $-36x+18{x}^{2}$.
The expression now becomes:
$\frac{48\left(x-2\right)}{-36x+18{x}^{2}}$.
To further simplify this expression, we can factor out a common factor of 6 from both the numerator and the denominator:
$\frac{6·8\left(x-2\right)}{6·\left(-6x+3{x}^{2}\right)}$.
Simplifying, we have:
$\frac{8\left(x-2\right)}{-6x+3{x}^{2}}$.
Finally, we can factor out a common factor of $-1$ from the numerator to get:
$\frac{-8\left(x-2\right)}{3{x}^{2}-6x}$.
Now, let's factor out a common factor of $x$ from the denominator:
$\frac{-8\left(x-2\right)}{3x\left(x-2\right)}$.
Notice that the term $\left(x-2\right)$ appears in both the numerator and denominator. We can cancel out this common factor:
$\frac{-8}{3x}$.
Therefore, the final simplified expression is:
$\frac{8}{-3x}$.
But we can further simplify this by multiplying both the numerator and denominator by $-1$ to get the positive sign in the numerator:
$\frac{-8}{-3x}$.
Simplifying the negative signs, we have:
$\frac{8}{3x}$.
So, the final answer in factored form is:
$\frac{8}{3x}$.

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