Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. (√7 - 1) / (2√7 + 4√2)

Verified step by step guidance

1. Identify the conjugate of the denominator, which is \(2\sqrt{7} - 4\sqrt{2}\).
2. Multiply both the numerator and the denominator by the conjugate \(2\sqrt{7} - 4\sqrt{2}\).
3. Apply the distributive property to expand the numerator: \((\sqrt{7} - 1)(2\sqrt{7} - 4\sqrt{2})\).
4. Use the difference of squares formula to simplify the denominator: \((2\sqrt{7})^2 - (4\sqrt{2})^2\).
5. Simplify the expression by combining like terms in the numerator and calculating the simplified denominator.

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 roots and simplifies the expression. This technique is essential in rationalizing denominators that contain two terms.

Properties of Square Roots

Understanding the properties of square roots is crucial for manipulating expressions involving them. Key properties include that √a * √b = √(ab) and that √(a/b) = √a / √b. These properties allow for the simplification of expressions and are particularly useful when rationalizing denominators, as they help in combining and simplifying terms effectively.