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Exercises Expressions, exponents and radicals

Example 6 from Algebra and Trigonometry, 1.6 Rational Expressions

Simplifying Complex Rational Expressions

Simplify: $\frac{y + \frac{1}{x}}{\frac{x}{y}}$ .

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
Begin by combining the expressions in the numerator into one expression.

$\begin{matrix} y \cdot \frac{x}{x} + \frac{1}{x} & Multiply by \frac{x}{x} to get LCD as denominator . \\ \frac{x y}{x} + \frac{1}{x} \\ \frac{x y + 1}{x} & Add numerators . \end{matrix}$
Now the numerator is a single rational expression and the denominator is a single rational expression.

$\frac{\frac{x y + 1}{x}}{\frac{x}{y}}$

We can rewrite this as division, and then multiplication.

$\begin{matrix} \frac{x y + 1}{x} \div \frac{x}{y} \\ \frac{x y + 1}{x} \cdot \frac{y}{x} & Rewrite as multiplication . \\ \frac{y (x y + 1)}{x^{2}} & Multiply . \end{matrix}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Simplify: $\frac{\frac{x}{y} - \frac{y}{x}}{y}$

How did it go?

PreviousSubtracting Rational Expressions

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 1.6, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.