Simplify each complex fraction. See Examples 5 and 6. (y/r) ÷ (x/y)

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Identify the complex fraction given: \( \frac{\frac{y}{r}}{\frac{x}{r}} \). This means you have a fraction divided by another fraction.

Recall that dividing by a fraction is equivalent to multiplying by its reciprocal. So rewrite the expression as \( \frac{y}{r} \times \frac{r}{x} \).

Multiply the numerators together and the denominators together: \( \frac{y \times r}{r \times x} \).

Notice that \( r \) appears in both numerator and denominator, so you can simplify by canceling \( r \) out: \( \frac{y \times \cancel{r}}{\cancel{r} \times x} = \frac{y}{x} \).

The simplified form of the complex fraction is \( \frac{y}{x} \).

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Complex Fractions

A complex fraction is a fraction where the numerator, denominator, or both contain fractions themselves. Simplifying involves rewriting the expression as a single fraction by eliminating the smaller fractions within it.

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Properties of Exponents

When variables have exponents, rules such as dividing powers with the same base (subtracting exponents) help simplify expressions. For example, \( \frac{x^a}{x^b} = x^{a-b} \) is essential in reducing terms in complex fractions.

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Dividing fractions involves multiplying by the reciprocal. To simplify complex fractions, convert division into multiplication by flipping the denominator fraction, then multiply numerators and denominators accordingly.

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