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

Verified step by step guidance

1. 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.
2. 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.
3. Multiply both the numerator and the denominator by the conjugate \(3 + \sqrt{r}\) to keep the expression equivalent: \(\frac{9 - r}{3 - \sqrt{r}} \times \frac{3 + \sqrt{r}}{3 + \sqrt{r}}\).
4. 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 \(3^2 - (\sqrt{r})^2\).
5. 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.

Key Concepts

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.