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Perform the indicated operation and simplify the result. Leave your answer in factored form.

$ [\dfrac {4x-8}{-3x}] .[\dfrac {12}{12-6x}] $

This question aims to simplify a fraction in its simplest form. A rational expression is reduced to the lowest terms if the numerator and denominator have no common factors.

Steps to simplify the fraction:

Step 1: Factor the numerator and denominator.

Step 2: List restricted values.

Step 3: Cancel the common factor.

Step 4: Reduce to the lowest terms and note any bounds not implied by the expression.

Expert Answer

Step 1

We can simplify algebraic expressions by performing the mathematical operation stated in it, removing common factors, and solving the equations to obtain a more straightforward form. Multiplying an algebraic expression is the same as multiplying fractions or rational functions. To perform multiplication between two algebraic expressions, we must multiply the numerator of the first algebraic expression by the numerator of the second expression and multiply the denominator of the first algebraic expression by the second algebraic expression.

Step 2

First, we can simplify by taking the common factors of the terms of the expression. Numerator $ 4x – 8 $ of the first fraction is a multiple of $ 4 $, it can be written as taking $ 4 $ outside the braces as $ 4 ( x – 2 ) $. The denominator $ 12 – 6x $ of the second fraction is a multiple of $ 6 $; it can be written as by taking $ 6 $ out of as $ 6(2 -x)$.

The expression can be written as

\[ \dfrac {4(x-2)}{-3x} \times \dfrac{12}{6(2-x)} \]

Now we can simplify the terms by canceling the multiples using the numerator and denominator.

\[ \dfrac {4 (x-2) }{-3x} \times \dfrac {12}{6(2-x)} = \dfrac { 4 (x-2) }{ -3x } \times \dfrac {2}{2-x} \]

\[ = \dfrac {8(x-2) }{ -3x (2 – x) } \]

$ (2-x) $ can be wrriten as $ -(x-2) $

\[ \dfrac { 8 (x-2) }{ -3x \times -(x-2)} = \dfrac{ 8 }{ 3x }  \]

Hence, the simplest factor is $\dfrac {8}{3x} $

Numerical Result

The simplest form of expression is $ [\dfrac { 4x – 8 }{ -3x }] .[\dfrac { 12 }{ 12 – 6x } ] $ is $\dfrac { 8 }{ 3x } $.

Example

Perform the given operation and simplify the result. Leave your answer in edited form.

$ ( \dfrac {x ^ {2} –  3x }{x ^ {2} – 5x } )$

Solution

Step 1: Factor the numerator and denominator.

\[ ( \dfrac {x ^ {2} –  3x }{x ^ {2} – 5x} ) = \dfrac { x (x-3) } {x (x-5) } \]

Step 2: List restricted values.

Here notice any restriction on $ x $. As division by $0 $ is undefined. Here we see that $ x \neq 0 $ and $ x \neq -5 $.

\[\dfrac { x ( x – 3) }{ x (x – 5) }\]

Step 3: Cancel the common factor.

Now notice that the numerator and denominator have a common factor of $ x $. This can be canceled.

\[ = \dfrac { x – 3 }{ x – 5 }\]

Hence, the simplest form is $\dfrac { x – 3 }{ x – 5 } $.

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