QuestionRationalize the denominator of $\frac{4}{3-\sqrt{7}}$ .

Studdy Solution

STEP 1

Assumptions1. We are given the fraction $\frac{4}{3-\sqrt{7}}$ . . We need to rationalize the denominator, which means we want to eliminate the square root from the denominator.

STEP 2

To rationalize the denominator, we need to multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of a binomial $(a - b)$ is $(a + b)$ . So, the conjugate of $( - \sqrt{7})$ is $( + \sqrt{7})$ .

STEP 3

Multiply the numerator and the denominator by the conjugate of the denominator.
$\frac{}{3-\sqrt{7}} \times \frac{3 + \sqrt{7}}{3 + \sqrt{7}}$

STEP 4

Apply the distributive property in the numerator and use the difference of squares formula in the denominator. The difference of squares formula states that for any real numbers a and b, $(a - b)(a + b) = a^2 - b^2$ .
$\frac{4 \times (3 + \sqrt{7})}{(3)^2 - (\sqrt{7})^2}$

STEP 5

implify the numerator and the denominator.
$\frac{12 +4\sqrt{7}}{9 -7}$

STEP 6

implify the fraction further.
$\frac{12 +4\sqrt{}}{2}$

STEP 7

Finally, divide each term in the numerator by the denominator to get the final answer.
$6 +2\sqrt{7}$ So, the rationalized form of $\frac{4}{3-\sqrt{7}}$ is $6 +2\sqrt{7}$ .

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