Rationalize the denominator.
$\(\frac{3}{3+\sqrt7}\)$

Verified step by step guidance

1. Identify the expression to rationalize: \(\frac{3}{3+\sqrt{7}}\) where the denominator contains a sum with a square root.
2. Multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(3 + \sqrt{7}\) is \(3 - \sqrt{7}\). So multiply by \(\frac{3 - \sqrt{7}}{3 - \sqrt{7}}\).
3. Apply the multiplication: The numerator becomes \(3 \times (3 - \sqrt{7})\) and the denominator becomes \((3 + \sqrt{7})(3 - \sqrt{7})\).
4. Use the difference of squares formula for the denominator: \((a + b)(a - b) = a^2 - b^2\). Here, \(a = 3\) and \(b = \sqrt{7}\), so the denominator simplifies to \(3^2 - (\sqrt{7})^2\).
5. Simplify the denominator by calculating \$3^2 = 9\( and \((\sqrt{7})^2 = 7\), so the denominator becomes \)9 - 7$. The numerator remains as \(3(3 - \sqrt{7})\).

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any radicals (square roots) 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 expression.

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, which removes the square root terms. This property is essential for rationalizing denominators containing sums or differences involving square roots.

Difference of Squares Formula

The difference of squares formula states that (x + y)(x - y) = x² - y². When applied to conjugates, it helps eliminate radicals by turning expressions like (3 + √7)(3 - √7) into 3² - (√7)² = 9 - 7 = 2, a rational number. This simplification is key to rationalizing denominators.