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

Verified step by step guidance

1. Identify the complex fraction given: \( \frac{\frac{y}{r}}{\frac{x}{r}} \). This means you have a fraction divided by another fraction.
2. Recall that dividing by a fraction is equivalent to multiplying by its reciprocal. So rewrite the expression as \( \frac{y}{r} \times \frac{r}{x} \).
3. Multiply the numerators together and the denominators together: \( \frac{y \times r}{r \times x} \).
4. Notice that \( r \) appears in both numerator and denominator, so you can simplify by canceling \( r \) out: \( \frac{y \times \cancel{r}}{\cancel{r} \times x} = \frac{y}{x} \).
5. The simplified form of the complex fraction is \( \frac{y}{x} \).

Key Concepts

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.

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.

Fraction Division and Multiplication

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.