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: \(\frac{p - 4}{\sqrt{p} + 2}\). The goal is to eliminate the square root from the denominator.
2. Multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(\sqrt{p} + 2\) is \(\sqrt{p} - 2\). So multiply by \(\frac{\sqrt{p} - 2}{\sqrt{p} - 2}\).
3. Apply the multiplication: The numerator becomes \((p - 4)(\sqrt{p} - 2)\) and the denominator becomes \((\sqrt{p} + 2)(\sqrt{p} - 2)\).
4. Use the difference of squares formula for the denominator: \((a + b)(a - b) = a^2 - b^2\). Here, \(a = \sqrt{p}\) and \(b = 2\), so the denominator simplifies to \(p - 4\).
5. Write the new expression as \(\frac{(p - 4)(\sqrt{p} - 2)}{p - 4}\). Since \(p - 4\) is common in numerator and denominator, consider simplifying the expression further if 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 to simplify the expression and make it easier to work with. Typically, this is achieved by multiplying the numerator and denominator by a conjugate or an appropriate radical.

Conjugates of Binomials

The conjugate of a binomial expression like (√p + 2) is (√p - 2). Multiplying a binomial by its conjugate results in a difference of squares, which removes the square root terms. This technique is essential for rationalizing denominators containing sums or differences involving radicals.

Properties of Square Roots and Nonnegative Variables

Since variables represent nonnegative numbers, the square root function √p is defined and nonnegative. This ensures that expressions involving √p behave predictably, allowing simplification without considering complex or negative values. Understanding this helps avoid extraneous solutions or undefined expressions.