QuestionRationalize the denominator of $\frac{6}{3+\sqrt{7}}$ . What is the simplified result?

Studdy Solution

STEP 1

Assumptions1. We are given a rational expression $\frac{6}{3+\sqrt{7}}$ . The task is to rationalize the denominator, which means to eliminate the square root from the denominator

STEP 2

To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a binomial $a + b$ is $a - b$ . So, the conjugate of $+ \sqrt{7}$ is $- \sqrt{7}$ .
$\frac{6}{+\sqrt{7}} \times \frac{-\sqrt{7}}{-\sqrt{7}}$

STEP 3

Multiply the numerators together and the denominators together.
$\frac{6 \times (3-\sqrt{7})}{(3+\sqrt{7}) \times (3-\sqrt{7})}$

STEP 4

istribute the6 in the numerator.
$\frac{18 -6\sqrt{7}}{(3+\sqrt{7}) \times (3-\sqrt{7})}$

STEP 5

In the denominator, use the difference of squares formula, which states that $(a - b)(a + b) = a^2 - b^2$ .
$\frac{18 -\sqrt{7}}{3^2 - (\sqrt{7})^2}$

STEP 6

implify the denominator.
$\frac{18 -6\sqrt{}}{9 -}$

STEP 7

implify the denominator further.
$\frac{18 -6\sqrt{7}}{2}$

STEP 8

Finally, divide each term in the numerator by the denominator.
$\frac{18}{2} - \frac{6\sqrt{7}}{2}$

STEP 9

implify each term.
$9 -3\sqrt{7}$ So, the rationalized form of $\frac{6}{3+\sqrt{7}}$ is $9 -3\sqrt{7}$ .

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