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: \( \frac{9 - r}{3 - \sqrt{r}} \).
2. Recognize that the denominator \(3 - \sqrt{r}\) is in the form of \(a - \sqrt{b}\).
3. To rationalize, multiply both the numerator and the denominator by the conjugate of the denominator, which is \(3 + \sqrt{r}\).
4. Set up the multiplication: \( \frac{(9 - r)(3 + \sqrt{r})}{(3 - \sqrt{r})(3 + \sqrt{r})} \).
5. Simplify the denominator using the difference of squares formula: \((3)^2 - (\sqrt{r})^2 = 9 - r\).

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any irrational numbers from the denominator of a fraction. This is typically achieved by multiplying both the numerator and the denominator by a suitable expression that will result in a rational number in the denominator. For example, if the denominator contains a square root, multiplying by the conjugate can help achieve this.

Conjugates

The conjugate of a binomial expression is formed by changing the sign between the two terms. For instance, the conjugate of (a + b) is (a - b). When multiplying a binomial by its conjugate, the result is a difference of squares, which eliminates the square root in the denominator, making it easier to simplify the expression.

Properties of Exponents and Radicals

Understanding the properties of exponents and radicals is crucial for manipulating expressions involving roots. For example, the square root of a product can be expressed as the product of the square roots, and the square of a square root returns the original value. This knowledge aids in simplifying expressions and rationalizing denominators effectively.