$ [\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 c**anceling 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 } $.