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Given below are the steps for dividing rational expressions:

**Step 1:** Factorize the rational expression, if possible.

**Step 2:** Take reciprocal of the second expression.

**Step 3:** Follow the procedure of multiplication of rational expressions.

**Step 4:** Reduce it in lowest form. A rational expression has been reduced to it's lowest term only when all the common factors from the numerator and the denominator has been cancelled.

Let us solve $\frac{12x^4}{x - 2}$ $\div $ $\frac{8x^2}{x - 2}$

Given expression is $\frac{12x^4}{x - 2}$ $\div $ $\frac{8x^2}{x - 2}$

Step 1: Change $\div $ sign into * sign.

$\frac{12x^4}{x - 2}$ * $\frac{x - 2}{8x^2}$

Step 2:Multiply numerators and denominators separately:

$\frac{(12x^4)(x - 2)}{8x^2(x - 2)}$

Step 3: Reduce it in the simplest form. Common factors from the numerator and the denominator can be cancelled.

$\frac{(3x^2)}{2}$

Below you can see how to divide Rational Expressions with some examples.### Solved Examples

**Question 1: **Simplify the following expressions

$\frac{56(x + 28)}{y}$ $\div$ $\frac{x + 28}{y}$

** Solution: **
**Question 2: **Solve the following expression:

$\frac{x^2 + 2x - 15}{x^2 - 4x - 45}$$\div$ $\frac{x^2 + x - 12}{x^2 - 5x - 36}$

** Solution: **
**Question 3: **Simplify the following expressions

$\frac{11p}{p + 4}$ $\div$ $\frac{p + 4}{p}$

** Solution: **

Let us solve $\frac{12x^4}{x - 2}$ $\div $ $\frac{8x^2}{x - 2}$

Step 1:

$\frac{12x^4}{x - 2}$ * $\frac{x - 2}{8x^2}$

Step 2:

$\frac{(12x^4)(x - 2)}{8x^2(x - 2)}$

Step 3:

$\frac{(3x^2)}{2}$

Below you can see how to divide Rational Expressions with some examples.

$\frac{56(x + 28)}{y}$ $\div$ $\frac{x + 28}{y}$

Given $\frac{56(x + 28)}{y}$ $\div$ $\frac{x + 28}{y}$

Step 1: Reciprocal the values and cross multiply

= $\frac{56(x + 28)}{y}$ $\times$ $\frac{y}{x + 28}$

**Step 2: **Cancelling common terms, we have

= $\frac{56(x + 28)}{y}$ $\times$ $\frac{y}{x + 28}$

= 56

Step 1:

= $\frac{56(x + 28)}{y}$ $\times$ $\frac{y}{x + 28}$

= $\frac{56(x + 28)}{y}$ $\times$ $\frac{y}{x + 28}$

= 56

$\frac{x^2 + 2x - 15}{x^2 - 4x - 45}$$\div$ $\frac{x^2 + x - 12}{x^2 - 5x - 36}$

We need to factor the numerators and the denominators of both the rational expressions.

The factors of the first rational expression are

Numerator = x^{2} + 2x - 15 = (x + 5)(x - 3)

Denominator = x^{2 }- 4x - 45 = (x - 9)(x + 5)

The factors of the second rational expression are

Numerator = x^{2} + x - 12 = (x + 4)(x - 3)

Denominator = x^{2 }- 5x - 36 = (x - 9) (x + 4)

Now $\frac{x^2 + 2x - 15}{x^2 - 4x - 45}$$\div$ $\frac{x^2 + x - 12}{x^2 - 5x - 36}$ has been reduced to $\frac{(x+5)(x-3)}{(x-9)(x+5)}$ $\div$ $\frac{(x+4)(x-3)}{ (x-9)(x+4)}$

= $\frac{(x+5)(x-3)}{(x-9)(x+5)}$ * $\frac{(x-9)(x+4)}{(x+4)(x-3)}$

Cancel the common terms

= 1

$\frac{11p}{p + 4}$ $\div$ $\frac{p + 4}{p}$

Given $\frac{11p}{p + 4}$ $\div$ $\frac{p + 4}{p}$**Step 1:** Reciprocal the values and cross multiply

= $\frac{11p}{p + 4}$ $\times$ $\frac{p}{p + 4}$**Step 2:** Cancelling common terms, we have

= $\frac{11p}{p + 4}$ $\times$ $\frac{p}{p + 4}$

= $\frac{11 p^2}{(p + 4)^2}$

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