QuestionSimplify: $\frac{6}{5-\sqrt{7}}$ . Choose the correct answer: a. $\frac{5+\sqrt{7}}{3}$ b. $\frac{5+\sqrt{7}}{6}$ c. $\frac{5+\sqrt{7}}{18}$ d. $15+3 \sqrt{7}$ .

Studdy Solution

STEP 1

Assumptions1. We are given the expression $\frac{6}{5-\sqrt{7}}$ and we need to simplify it completely.

STEP 2

To simplify this expression, we need to rationalize the denominator. We do this by multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of a binomial expression $a - b$ is $a + b$ . So, the conjugate of $5 - \sqrt{7}$ is $5 + \sqrt{7}$ .
$\frac{6}{5-\sqrt{7}} \times \frac{5 + \sqrt{7}}{5 + \sqrt{7}}$

STEP 3

Now, we multiply the numerator and the denominator separately.
$\frac{6(5 + \sqrt{7})}{(5-\sqrt{7})(5 + \sqrt{7})}$

STEP 4

We simplify the numerator and the denominator separately. In the denominator, we use the formula $(a-b)(a+b) = a^2 - b^2$ .
$\frac{30 +6\sqrt{7}}{^2 - (\sqrt{7})^2}$

STEP 5

implify the denominator.
$\frac{30 +\sqrt{7}}{25 -7}$

STEP 6

implify the denominator.
$\frac{30 +6\sqrt{}}{18}$

STEP 7

We can simplify this further by dividing each term in the numerator by6.
$\frac{5 + \sqrt{7}}{3}$ So, the simplified form of the given expression is $\frac{5 + \sqrt{7}}{3}$ , which corresponds to option (a).

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QuestionSimplify: $\frac{6}{5-\sqrt{7}}$ . Choose the correct answer: a. $\frac{5+\sqrt{7}}{3}$ b. $\frac{5+\sqrt{7}}{6}$ c. $\frac{5+\sqrt{7}}{18}$ d. $15+3 \sqrt{7}$ .

Studdy Solution

STEP 1

Assumptions1. We are given the expression $\frac{6}{5-\sqrt{7}}$ and we need to simplify it completely.

STEP 2

STEP 3

Now, we multiply the numerator and the denominator separately.
$\frac{6(5 + \sqrt{7})}{(5-\sqrt{7})(5 + \sqrt{7})}$

STEP 4

We simplify the numerator and the denominator separately. In the denominator, we use the formula $(a-b)(a+b) = a^2 - b^2$ .
$\frac{30 +6\sqrt{7}}{^2 - (\sqrt{7})^2}$

STEP 5

implify the denominator.
$\frac{30 +\sqrt{7}}{25 -7}$

STEP 6

implify the denominator.
$\frac{30 +6\sqrt{}}{18}$

STEP 7

We can simplify this further by dividing each term in the numerator by6.
$\frac{5 + \sqrt{7}}{3}$ So, the simplified form of the given expression is $\frac{5 + \sqrt{7}}{3}$ , which corresponds to option (a).

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QuestionSimplify: 65−7\frac{6}{5-\sqrt{7}}5−7​6​. Choose the correct answer: a. 5+73\frac{5+\sqrt{7}}{3}35+7​​ b. 5+76\frac{5+\sqrt{7}}{6}65+7​​ c. 5+718\frac{5+\sqrt{7}}{18}185+7​​ d. 15+3715+3 \sqrt{7}15+37​.

Studdy Solution

STEP 1

Assumptions1. We are given the expression 65−7\frac{6}{5-\sqrt{7}}5−7​6​ and we need to simplify it completely.

STEP 2

STEP 3

Now, we multiply the numerator and the denominator separately.6(5+7)(5−7)(5+7)\frac{6(5 + \sqrt{7})}{(5-\sqrt{7})(5 + \sqrt{7})}(5−7​)(5+7​)6(5+7​)​

STEP 4

We simplify the numerator and the denominator separately. In the denominator, we use the formula (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2(a−b)(a+b)=a2−b2.30+672−(7)2\frac{30 +6\sqrt{7}}{^2 - (\sqrt{7})^2}2−(7​)230+67​​

STEP 5

implify the denominator.30+725−7\frac{30 +\sqrt{7}}{25 -7}25−730+7​​

STEP 6

implify the denominator.30+618\frac{30 +6\sqrt{}}{18}1830+6​​

STEP 7

We can simplify this further by dividing each term in the numerator by6.5+73\frac{5 + \sqrt{7}}{3}35+7​​So, the simplified form of the given expression is 5+73\frac{5 + \sqrt{7}}{3}35+7​​, which corresponds to option (a).

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QuestionSimplify: $\frac{6}{5-\sqrt{7}}$ . Choose the correct answer: a. $\frac{5+\sqrt{7}}{3}$ b. $\frac{5+\sqrt{7}}{6}$ c. $\frac{5+\sqrt{7}}{18}$ d. $15+3 \sqrt{7}$ .

Assumptions1. We are given the expression $\frac{6}{5-\sqrt{7}}$ and we need to simplify it completely.

Now, we multiply the numerator and the denominator separately.
$\frac{6(5 + \sqrt{7})}{(5-\sqrt{7})(5 + \sqrt{7})}$

We simplify the numerator and the denominator separately. In the denominator, we use the formula $(a-b)(a+b) = a^2 - b^2$ .
$\frac{30 +6\sqrt{7}}{^2 - (\sqrt{7})^2}$

implify the denominator.
$\frac{30 +\sqrt{7}}{25 -7}$

implify the denominator.
$\frac{30 +6\sqrt{}}{18}$

We can simplify this further by dividing each term in the numerator by6.
$\frac{5 + \sqrt{7}}{3}$ So, the simplified form of the given expression is $\frac{5 + \sqrt{7}}{3}$ , which corresponds to option (a).