Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. $\frac{\sqrt{7} - 1}{2\sqrt{7} + 4\sqrt{2}}$

Verified step by step guidance

1. Identify the denominator: \$2\sqrt{7} + 4\sqrt{2}$. To rationalize it, we need to eliminate the square roots from the denominator.
2. Notice that the denominator is a sum of two terms involving square roots. To rationalize, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of \$2\sqrt{7} + 4\sqrt{2}\( is \)2\sqrt{7} - 4\sqrt{2}$.
3. Multiply both numerator and denominator by the conjugate: \(\frac{\sqrt{7} - 1}{2\sqrt{7} + 4\sqrt{2}} \times \frac{2\sqrt{7} - 4\sqrt{2}}{2\sqrt{7} - 4\sqrt{2}}\).
4. Use the difference of squares formula for the denominator: \((a + b)(a - b) = a^2 - b^2\). Here, \(a = 2\sqrt{7}\) and \(b = 4\sqrt{2}\). Calculate \(a^2\) and \(b^2\) separately.
5. Expand the numerator by distributing \((\sqrt{7} - 1)\) with \((2\sqrt{7} - 4\sqrt{2})\) using the distributive property (FOIL method). Simplify the resulting expression.

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves rewriting a fraction so that its denominator contains no radicals. This is done by multiplying the numerator and denominator by a suitable expression, often the conjugate, to eliminate the square roots from the denominator.

Conjugates of Binomials

The conjugate of a binomial expression a + b is a - b, and vice versa. Multiplying a binomial by its conjugate results in a difference of squares, which helps remove radicals from denominators by producing rational numbers.

Properties of Square Roots and Nonnegative Variables

Square roots represent nonnegative values, and when variables are nonnegative, simplifications involving radicals are straightforward. This assumption ensures no ambiguity in sign when rationalizing and simplifies the manipulation of expressions with radicals.