Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. (9 - r) / (3 - √r)

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Identify the expression to rationalize: \(\frac{9 - r}{3 - \sqrt{r}}\). The denominator contains a square root, so we want to eliminate it by multiplying by the conjugate.

Write the conjugate of the denominator \(3 - \sqrt{r}\), which is \(3 + \sqrt{r}\). Multiplying by this conjugate will help remove the square root from the denominator.

Multiply both the numerator and the denominator by the conjugate \(3 + \sqrt{r}\) to keep the expression equivalent: \(\frac{9 - r}{3 - \sqrt{r}} \times \frac{3 + \sqrt{r}}{3 + \sqrt{r}}\).

Use the difference of squares formula for the denominator: \((a - b)(a + b) = a^2 - b^2\). Here, \(a = 3\) and \(b = \sqrt{r}\), so the denominator becomes \(3^2 - (\sqrt{r})^2\).

Expand the numerator by distributing \((9 - r)(3 + \sqrt{r})\) using the distributive property (FOIL method), then simplify both numerator and denominator as much as possible.

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주요 개념

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Rationalizing the Denominator

Rationalizing the denominator involves eliminating any square roots or irrational numbers from the denominator of a fraction. This is done by multiplying the numerator and denominator by a conjugate or an appropriate expression to create a rational denominator.

Conjugates of Binomials

The conjugate of a binomial expression like (a - √b) is (a + √b). Multiplying a binomial by its conjugate results in a difference of squares, which removes the square root and simplifies the denominator to a rational number.

Properties of Square Roots and Nonnegative Variables

Since variables represent nonnegative numbers, square roots are defined and real. This ensures that expressions like √r are valid and simplifies the process of rationalization without considering complex numbers.