Rationalize the denominator. $\frac{7}{\sqrt{5} - 2}$

Verified step by step guidance

1. Identify the expression to rationalize: \(\frac{7}{\sqrt{5} - 2}\), where the denominator contains a radical.
2. To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(\sqrt{5} - 2\) is \(\sqrt{5} + 2\).
3. Set up the multiplication: \(\frac{7}{\sqrt{5} - 2} \times \frac{\sqrt{5} + 2}{\sqrt{5} + 2}\).
4. Multiply the numerators: \$7 \times (\sqrt{5} + 2)\(, and multiply the denominators using the difference of squares formula: \)(\sqrt{5})^2 - (2)^2$.
5. Simplify the denominator to \$5 - 4\(, and write the expression as \)\frac{7(\sqrt{5} + 2)}{1}$, which completes the rationalization process.

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any irrational numbers, such as square roots, from the denominator of a fraction. This is done to simplify the expression and make it easier to work with or interpret.

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 eliminate square roots in the denominator.

Difference of Squares Formula

The difference of squares formula states that (a + b)(a - b) = a² - b². This property is used when multiplying by the conjugate to remove radicals from the denominator by turning it into a rational number.