Rationalize each denominator. Assume all variables represent nonnegative numbers and that no denominators are 0. 1/(2 + √5)

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Identify the denominator that needs to be rationalized: \(2 + \sqrt{5}\). The goal is to eliminate the square root from the denominator.

Multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2 + \sqrt{5}\) is \(2 - \sqrt{5}\), so multiply by \(\frac{2 - \sqrt{5}}{2 - \sqrt{5}}\).

Apply the multiplication in the numerator: \(1 \times (2 - \sqrt{5}) = 2 - \sqrt{5}\).

Apply the multiplication in the denominator using the difference of squares formula: \((2 + \sqrt{5})(2 - \sqrt{5}) = 2^2 - (\sqrt{5})^2 = 4 - 5\).

Simplify the denominator expression \(4 - 5\) to get \(-1\), and write the rationalized expression as \(\frac{2 - \sqrt{5}}{-1}\). You can then simplify the fraction by dividing numerator and denominator by \(-1\).

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Rationalizing the Denominator

Rationalizing the denominator involves rewriting a fraction so that its denominator contains no radicals. This is done to simplify expressions and make them easier to work with. For denominators with square roots, multiplying numerator and denominator by a suitable expression removes the radical from the denominator.

Conjugates of Binomials

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

Properties of Square Roots and Nonnegative Variables

Square roots represent nonnegative values, and assuming variables are nonnegative ensures expressions remain defined and simplifies manipulation. This assumption allows us to avoid absolute value considerations when simplifying radicals and rationalizing denominators.