Math

QuestionRationalize the denominator of the expression 3+73+\sqrt{7}. What is the simplified result?

Studdy Solution

STEP 1

Assumptions1. We are given the expression 3+7\overline{3+\sqrt{7}} and asked to rationalize the denominator. . Rationalizing the denominator means to eliminate any square roots in the denominator of the fraction.

STEP 2

The given expression can be written as a fraction.
1+7\frac{1}{+\sqrt{7}}

STEP 3

To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a binomial a+ba + b is aba - b.
13+7×3737\frac{1}{3+\sqrt{7}} \times \frac{3-\sqrt{7}}{3-\sqrt{7}}

STEP 4

Multiply the numerators together and the denominators together.
1×(37)(3+7)×(37)\frac{1 \times (3-\sqrt{7})}{(3+\sqrt{7}) \times (3-\sqrt{7})}

STEP 5

implify the numerator.
37(3+7)×(37)\frac{3-\sqrt{7}}{(3+\sqrt{7}) \times (3-\sqrt{7})}

STEP 6

implify the denominator by using the formula (a+b)(ab)=a2b2(a+b)(a-b) = a^2 - b^2.
332()2\frac{3-\sqrt{}}{3^2 - (\sqrt{})^2}

STEP 7

implify the denominator further.
3797\frac{3-\sqrt{7}}{9 -7}

STEP 8

Calculate the denominator.
372\frac{3-\sqrt{7}}{2}So, the rationalized form of 3+7\overline{3+\sqrt{7}} is 372\frac{3-\sqrt{7}}{2}.

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