Solved on Feb 07, 2024

Write each expression in the form $a + b\sqrt{7}$ . (1) $\frac{3}{\sqrt{7}}$ , (2) $\frac{5 + \sqrt{7}}{3 - \sqrt{7}}$ .

STEP 1

Assumptions
1. We need to express each given expression in the form of $a + b\sqrt{7}$ .
2. Rationalizing the denominator is a common technique used when the denominator contains a square root.

STEP 2

Start with the first expression $\frac{3}{\sqrt{7}}$ . To write it in the form of $a + b\sqrt{7}$ , we need to rationalize the denominator.

STEP 3

Multiply the numerator and the denominator by $\sqrt{7}$ to rationalize the denominator.
$\frac{3}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}}$

STEP 4

Calculate the result of the multiplication.
$\frac{3\sqrt{7}}{7}$

STEP 5

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 1 in this case.
$\frac{3\sqrt{7}}{7} = \frac{3}{7}\sqrt{7}$

STEP 6

Express the result in the form of $a + b\sqrt{7}$ .
$\frac{3}{\sqrt{7}} = 0 + \frac{3}{7}\sqrt{7}$

STEP 7

Now, let's move on to the second expression $\frac{5+\sqrt{7}}{3-\sqrt{7}}$ . To write it in the form of $a + b\sqrt{7}$ , we will rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator.

STEP 8

The conjugate of $3 - \sqrt{7}$ is $3 + \sqrt{7}$ . Multiply the numerator and the denominator by the conjugate.
$\frac{5+\sqrt{7}}{3-\sqrt{7}} \times \frac{3+\sqrt{7}}{3+\sqrt{7}}$

STEP 9

Use the difference of squares formula to simplify the denominator:
$(a - b)(a + b) = a^2 - b^2$

STEP 10

Apply the formula to the denominator:
$\left(3 - \sqrt{7}\right)\left(3 + \sqrt{7}\right) = 3^2 - \left(\sqrt{7}\right)^2$

STEP 11

Calculate the denominator:
$3^2 - \left(\sqrt{7}\right)^2 = 9 - 7$

STEP 12

Simplify the denominator:
$9 - 7 = 2$

STEP 13

Expand the numerator by distributing:
$(5 + \sqrt{7})(3 + \sqrt{7}) = 5 \cdot 3 + 5\sqrt{7} + 3\sqrt{7} + \sqrt{7}\sqrt{7}$

STEP 14

Simplify the numerator:
$15 + 5\sqrt{7} + 3\sqrt{7} + 7$

STEP 15

Combine like terms in the numerator:
$15 + 7 + (5 + 3)\sqrt{7}$

STEP 16

Complete the simplification of the numerator:
$22 + 8\sqrt{7}$

STEP 17

Now, write the simplified expression with the rationalized denominator:
$\frac{22 + 8\sqrt{7}}{2}$

STEP 18

Divide both terms in the numerator by the denominator:
$\frac{22}{2} + \frac{8\sqrt{7}}{2}$

STEP 19

Simplify the terms:
$11 + 4\sqrt{7}$

STEP 20

Now we have both expressions in the desired form:
1. $\frac{3}{\sqrt{7}} = 0 + \frac{3}{7}\sqrt{7}$
2. $\frac{5+\sqrt{7}}{3-\sqrt{7}} = 11 + 4\sqrt{7}$

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Write each expression in the form a+b7a + b\sqrt{7}a+b7​. (1) 37\frac{3}{\sqrt{7}}7​3​, (2) 5+73−7\frac{5 + \sqrt{7}}{3 - \sqrt{7}}3−7​5+7​​.

STEP 1

Assumptions1. We need to express each given expression in the form of a+b7a + b\sqrt{7}a+b7​.2. Rationalizing the denominator is a common technique used when the denominator contains a square root.

STEP 2

Start with the first expression 37\frac{3}{\sqrt{7}}7​3​. To write it in the form of a+b7a + b\sqrt{7}a+b7​, we need to rationalize the denominator.

STEP 3

Multiply the numerator and the denominator by 7\sqrt{7}7​ to rationalize the denominator.37×77\frac{3}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}}7​3​×7​7​​

STEP 4

Calculate the result of the multiplication.377\frac{3\sqrt{7}}{7}737​​

STEP 5

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 1 in this case.377=377\frac{3\sqrt{7}}{7} = \frac{3}{7}\sqrt{7}737​​=73​7​

STEP 6

Express the result in the form of a+b7a + b\sqrt{7}a+b7​.37=0+377\frac{3}{\sqrt{7}} = 0 + \frac{3}{7}\sqrt{7}7​3​=0+73​7​

STEP 7

Now, let's move on to the second expression 5+73−7\frac{5+\sqrt{7}}{3-\sqrt{7}}3−7​5+7​​. To write it in the form of a+b7a + b\sqrt{7}a+b7​, we will rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator.

STEP 8

The conjugate of 3−73 - \sqrt{7}3−7​ is 3+73 + \sqrt{7}3+7​. Multiply the numerator and the denominator by the conjugate.5+73−7×3+73+7\frac{5+\sqrt{7}}{3-\sqrt{7}} \times \frac{3+\sqrt{7}}{3+\sqrt{7}}3−7​5+7​​×3+7​3+7​​

STEP 9

Use the difference of squares formula to simplify the denominator:(a−b)(a+b)=a2−b2(a - b)(a + b) = a^2 - b^2(a−b)(a+b)=a2−b2

STEP 10

Apply the formula to the denominator:(3−7)(3+7)=32−(7)2\left(3 - \sqrt{7}\right)\left(3 + \sqrt{7}\right) = 3^2 - \left(\sqrt{7}\right)^2(3−7​)(3+7​)=32−(7​)2

STEP 11

Calculate the denominator:32−(7)2=9−73^2 - \left(\sqrt{7}\right)^2 = 9 - 732−(7​)2=9−7

STEP 12

Simplify the denominator:9−7=29 - 7 = 29−7=2

STEP 13

Expand the numerator by distributing:(5+7)(3+7)=5⋅3+57+37+77(5 + \sqrt{7})(3 + \sqrt{7}) = 5 \cdot 3 + 5\sqrt{7} + 3\sqrt{7} + \sqrt{7}\sqrt{7}(5+7​)(3+7​)=5⋅3+57​+37​+7​7​

STEP 14

Simplify the numerator:15+57+37+715 + 5\sqrt{7} + 3\sqrt{7} + 715+57​+37​+7

STEP 15

Combine like terms in the numerator:15+7+(5+3)715 + 7 + (5 + 3)\sqrt{7}15+7+(5+3)7​

STEP 16

Complete the simplification of the numerator:22+8722 + 8\sqrt{7}22+87​

STEP 17

Now, write the simplified expression with the rationalized denominator:22+872\frac{22 + 8\sqrt{7}}{2}222+87​​

STEP 18

Divide both terms in the numerator by the denominator:222+872\frac{22}{2} + \frac{8\sqrt{7}}{2}222​+287​​

STEP 19

Simplify the terms:11+4711 + 4\sqrt{7}11+47​

STEP 20

Now we have both expressions in the desired form:1. 37=0+377\frac{3}{\sqrt{7}} = 0 + \frac{3}{7}\sqrt{7}7​3​=0+73​7​2. 5+73−7=11+47\frac{5+\sqrt{7}}{3-\sqrt{7}} = 11 + 4\sqrt{7}3−7​5+7​​=11+47​

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Write each expression in the form $a + b\sqrt{7}$ . (1) $\frac{3}{\sqrt{7}}$ , (2) $\frac{5 + \sqrt{7}}{3 - \sqrt{7}}$ .

Assumptions
1. We need to express each given expression in the form of $a + b\sqrt{7}$ .
2. Rationalizing the denominator is a common technique used when the denominator contains a square root.

Start with the first expression $\frac{3}{\sqrt{7}}$ . To write it in the form of $a + b\sqrt{7}$ , we need to rationalize the denominator.

Multiply the numerator and the denominator by $\sqrt{7}$ to rationalize the denominator.
$\frac{3}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}}$

Calculate the result of the multiplication.
$\frac{3\sqrt{7}}{7}$

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 1 in this case.
$\frac{3\sqrt{7}}{7} = \frac{3}{7}\sqrt{7}$

Express the result in the form of $a + b\sqrt{7}$ .
$\frac{3}{\sqrt{7}} = 0 + \frac{3}{7}\sqrt{7}$

Now, let's move on to the second expression $\frac{5+\sqrt{7}}{3-\sqrt{7}}$ . To write it in the form of $a + b\sqrt{7}$ , we will rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator.

The conjugate of $3 - \sqrt{7}$ is $3 + \sqrt{7}$ . Multiply the numerator and the denominator by the conjugate.
$\frac{5+\sqrt{7}}{3-\sqrt{7}} \times \frac{3+\sqrt{7}}{3+\sqrt{7}}$

Use the difference of squares formula to simplify the denominator:
$(a - b)(a + b) = a^2 - b^2$

Apply the formula to the denominator:
$\left(3 - \sqrt{7}\right)\left(3 + \sqrt{7}\right) = 3^2 - \left(\sqrt{7}\right)^2$

Calculate the denominator:
$3^2 - \left(\sqrt{7}\right)^2 = 9 - 7$

Simplify the denominator:
$9 - 7 = 2$

Expand the numerator by distributing:
$(5 + \sqrt{7})(3 + \sqrt{7}) = 5 \cdot 3 + 5\sqrt{7} + 3\sqrt{7} + \sqrt{7}\sqrt{7}$

Simplify the numerator:
$15 + 5\sqrt{7} + 3\sqrt{7} + 7$

Combine like terms in the numerator:
$15 + 7 + (5 + 3)\sqrt{7}$

Complete the simplification of the numerator:
$22 + 8\sqrt{7}$

Now, write the simplified expression with the rationalized denominator:
$\frac{22 + 8\sqrt{7}}{2}$

Divide both terms in the numerator by the denominator:
$\frac{22}{2} + \frac{8\sqrt{7}}{2}$

Simplify the terms:
$11 + 4\sqrt{7}$

Now we have both expressions in the desired form:
1. $\frac{3}{\sqrt{7}} = 0 + \frac{3}{7}\sqrt{7}$
2. $\frac{5+\sqrt{7}}{3-\sqrt{7}} = 11 + 4\sqrt{7}$