Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. (p - 4) / (√p + 2)

Verified step by step guidance

1. Identify the expression to rationalize: \( \frac{p - 4}{\sqrt{p} + 2} \).
2. To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{p} - 2 \).
3. Write the expression as: \( \frac{(p - 4)(\sqrt{p} - 2)}{(\sqrt{p} + 2)(\sqrt{p} - 2)} \).
4. Simplify the denominator using the difference of squares formula: \((\sqrt{p})^2 - 2^2 = p - 4\).
5. Expand the numerator: \((p - 4)(\sqrt{p} - 2)\) and simplify if possible.

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 is a square root, multiplying by the same square root 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 (√p + 2) is (√p - 2). When multiplying a binomial by its conjugate, the result is a difference of squares, which simplifies the expression and is particularly useful in rationalizing denominators that contain square roots.

Properties of Exponents and Radicals

Understanding the properties of exponents and radicals is essential for manipulating expressions involving roots. Key properties include that the square root of a product is the product of the square roots, and that raising a power to a power involves multiplying the exponents. These properties help simplify expressions and are crucial when working with rationalization and simplification of algebraic fractions.