Simplifying Rational Expressions

A rational expression is an algebraic expression of the form $\frac{A}{B}$, where $A$ and $B$ are polynomials and $B \neq 0$.

Given the polynomials $A, B, C,$ and $D,$ where $B\neq0$ and $D\neq0,$ we can perform the following operations:

(1) Addition: ${\frac { A }{ B } + \frac { C }{ B } =\frac { A + C }{ B },~~ \frac { A }{ B } + \frac { C }{ D } =\frac { AD + BC }{ BD }}$

(2) Subtraction: ${\frac { A }{ B } - \frac { C }{ B } =\frac { A - C }{ B },~~\frac { A }{ B } - \frac { C }{ D } =\frac { AD - BC }{ BD }}$

(3) Multiplication: ${\frac { A }{ B } \times \frac { C }{ D } =\frac { AC }{ BD }}$

(4) Division: ${\frac { A }{ B } \div \frac { C }{ D } =\frac{A}{B}\times\frac{D}{C} = \frac { AD }{ BC }}$ $($where $C\neq0)$

Add $\displaystyle{\frac{6x^3}{7x^5z^4}+\frac{2x}{3y^4z^2}}.$

---

We have

$$\frac{18y^4+14x^3z^2}{21x^2y^4z^4}. \ _\square$$

Subtract $\displaystyle{ \frac{3}{4ab^2} - \frac{2}{3a^2b}}.$

---

We have

$$\frac{9a-8b}{12a^2b^2}. \ _\square$$

Simplify $\displaystyle{ \frac{ 3 x }{ 2 y} \times \frac{ 4 xy }{ 5 x }}.$

---

We have

$$\frac{ 3 x }{ 2 y} \times \frac{ 4 xy }{ 5 x } = \frac { 12 x^2y }{ 10 xy } = \frac { 6x }{ 5 }. \ _\square$$

Simplify $\displaystyle{ \frac{ 3 x^2 y^2 }{ 4 x^3 y} \times \frac{ 4 x^3 y^4 }{ 5 x^2 }}.$

---

We have

$$\frac{ 3 x^2 y^2 }{ 4 x^3 y} \times \frac{ 4 x^3 y^4 }{ 5 x^2 } = \frac{12x^5y^6}{20x^5y} = \frac { 3y^5 }{ 5 }. \ _\square$$

Simplify $\displaystyle{ \frac{ x^2 - x - 2}{x^2 - 2x } \times \frac{ x }{ x - 3}}.$

---

We have

$$\begin{aligned} \frac{ x^2 - x - 2}{x^2 - 2x } \times \frac{ x }{ x - 3} &= \frac{ (x - 2)(x + 1) }{ x( x - 2 ) } \times \frac{ x }{ x - 3} \\ &= \frac{ x(x - 2)(x + 1) }{ x( x - 2)(x - 3)} \\ &= \frac{ x+1 }{ x - 3}. \ _\square \end{aligned}$$

Simplify $\displaystyle{ \frac{ 2x }{ 3y } \div \frac{ 5y }{ x }}.$

---

We have

$$\frac{ 2x }{ 3y } \div \frac{ 5y }{ x } = \frac{ 2x }{ 3y } \times \frac{ x }{ 5y } = \frac { 2x^2 }{ 15 y^2 }. \ _\square$$

Simplify $\displaystyle{ \frac{ 25x^2 y }{ 3 } \div \frac{ 5y }{ 2x }}.$

---

We have

$$\frac{ 25x^2 y }{ 3 } \div \frac{ 5y }{ 2x } = \frac{ 25x^2 y }{ 3 } \times \frac{ 2x }{ 5y } = \frac { 50 x^3 y }{ 15 y } = \frac { 50 x^3 }{ 15 } = \frac { 10 x^3 }{ 3 }. \ _\square$$

Simplify $\displaystyle{ \frac{ a^2 - b^2 }{ a + b } \div \frac{ a^2 - 2ab + b^2 }{ a^3 - b^3 } }.$

---

We have

\[ \begin{align} \frac{ a^2 - b^2 }{ a + b } \div \frac{ a^2 - 2ab + b^2 }{ a^3 - b^3 }
&= \frac{ a^2 - b^2 }{ a + b } \times \frac{ a^3 - b^3 }{ a^2 - 2ab + b^2 } \\ &= \frac{ (a-b)(a+b) }{ a + b } \times \frac{ (a-b)(a^2 + ab + b^2) }{ (a-b)^2 } \\ &= \frac{ (a-b)^2(a+b)(a^2 + ab + b^2) }{ (a+b)(a-b)^2 } \\ &= a^2 + ab + b^2 . \ _\square \end{align} \]

A rational expression is said to be reduced to the lowest term or simplest form if $1$ is the only common factor of its numerator and denominator. To reduce rational expressions, we factorize the numerator and denominator and then find their common factors. Then we cancel the common factors from both the numerator and denominator, which will give the simplest form.

Simplify $\displaystyle{ \frac { x+3 }{ { x }^{ 3 }+3{ x }^{ 2 } }}.$

---

We have

$$\begin{aligned} \frac { x+3 }{ { x }^{ 3 }+3{ x }^{ 2 } } &=\frac { x+3 }{ { x }^{ 2 }(x+3) }\qquad \qquad \big({ x }^{ 2 }(x+3)\neq 0 \text{ or } x\neq 0,-3\big)\\\\ &=\frac { 1 }{ { x }^{ 2 } }. \ _\square \end{aligned}$$

Simplify

$$\left ( \frac{x+7}{x^{2}-x-6} \right ) \div \left ( \frac{2}{x-3}+ \frac{-1}{x+2} \right ).$$

---

We have

$$\begin{aligned} \left ( \frac{x+7}{x^{2}-x-6} \right ) \div \left ( \frac{2}{x-3}+ \frac{-1}{x+2} \right ) &=\frac{x+7}{(x-3)(x+2)} \div \left ( \frac{2(x+2)-1(x-3)}{(x-3)(x+2)} \right )\\\\ &=\frac{x+7}{(x-3)(x+2)}\times \frac{(x-3)(x+2)}{x+7} \qquad \qquad (x \neq 3,-2,-7)\\\\ &=1. \ _\square \end{aligned}$$

For help simplifying complex fractions—fractions with a numerator or denominator that also contains a fraction—see the Complex Fractions page.