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Given a complex rational expression, simplify it.
Simplify: $\frac{y+\frac{1}{x}}{\frac{x}{y}}$ .
Begin by combining the expressions in the numerator into one expression.
Now the numerator is a single rational expression and the denominator is a single rational expression.
We can rewrite this as division, and then multiplication.
Simplify: $\frac{\frac{x}{y}-\frac{y}{x}}{y}$
$\frac{{x}^{2}-{y}^{2}}{x{y}^{2}}$
Can a complex rational expression always be simplified?
Yes. We can always rewrite a complex rational expression as a simplified rational expression.
Access these online resources for additional instruction and practice with rational expressions.
How can you use factoring to simplify rational expressions?
You can factor the numerator and denominator to see if any of the terms can cancel one another out.
How do you use the LCD to combine two rational expressions?
Tell whether the following statement is true or false and explain why: You only need to find the LCD when adding or subtracting rational expressions.
True. Multiplication and division do not require finding the LCD because the denominators can be combined through those operations, whereas addition and subtraction require like terms.
For the following exercises, simplify the rational expressions.
$\frac{{x}^{2}-16}{{x}^{2}-5x+4}$
$\frac{6{a}^{2}-24a+24}{6{a}^{2}-24}$
$\frac{m-12}{{m}^{2}-144}$
$\frac{6{x}^{2}+5x-4}{3{x}^{2}+19x+20}$
$\frac{3{c}^{2}+25c-18}{3{c}^{2}-23c+14}$
$\frac{12{n}^{2}-29n-8}{28{n}^{2}-5n-3}$
$\frac{3n-8}{7n-3}$
For the following exercises, multiply the rational expressions and express the product in simplest form.
$\frac{{x}^{2}-x-6}{2{x}^{2}+x-6}\cdot \frac{2{x}^{2}+7x-15}{{x}^{2}-9}$
$\frac{{c}^{2}+2c-24}{{c}^{2}+12c+36}\cdot \frac{{c}^{2}-10c+24}{{c}^{2}-8c+16}$
$\frac{c-6}{c+6}$
$\frac{2{d}^{2}+9d-35}{{d}^{2}+10d+21}\cdot \frac{3{d}^{2}+2d-21}{3{d}^{2}+14d-49}$
$\frac{10{h}^{2}-9h-9}{2{h}^{2}-19h+24}\cdot \frac{{h}^{2}-16h+64}{5{h}^{2}-37h-24}$
$1$
$\frac{6{b}^{2}+13b+6}{4{b}^{2}-9}\cdot \frac{6{b}^{2}+31b-30}{18{b}^{2}-3b-10}$
$\frac{2{d}^{2}+15d+25}{4{d}^{2}-25}\cdot \frac{2{d}^{2}-15d+25}{25{d}^{2}-1}$
$\frac{{d}^{2}-25}{25{d}^{2}-1}$
$\frac{6{x}^{2}-5x-50}{15{x}^{2}-44x-20}\cdot \frac{20{x}^{2}-7x-6}{2{x}^{2}+9x+10}$
$\frac{{t}^{2}-1}{{t}^{2}+4t+3}\cdot \frac{{t}^{2}+2t-15}{{t}^{2}-4t+3}$
$\frac{t+5}{t+3}$
$\frac{2{n}^{2}-n-15}{6{n}^{2}+13n-5}\cdot \frac{12{n}^{2}-13n+3}{4{n}^{2}-15n+9}$
$\frac{36{x}^{2}-25}{6{x}^{2}+65x+50}\cdot \frac{3{x}^{2}+32x+20}{18{x}^{2}+27x+10}$
$\frac{6x-5}{6x+5}$
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