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1.Simplify $\frac{6}{2\sqrt 3 - 3}$ as much as possible.

2.Simplify $\dfrac{1}{\sqrt2-1}+\dfrac{1}{\sqrt2+1}$ as much as possible.

3.Simplify $\frac{2\sqrt[3]9}{1 + \sqrt[3]3 + \sqrt[3]9}.$

Thank you in advance!

Guest Feb 2, 2018
 #1
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2.

 

Simplify the following:

1/(sqrt(2) - 1) + 1/(sqrt(2) + 1)

 

Multiply numerator and denominator of 1/(sqrt(2) - 1) by the conjugate of the denominator.

Multiply numerator and denominator of 1/(sqrt(2) - 1) by sqrt(2) + 1:

(sqrt(2) + 1)/((sqrt(2) - 1) (sqrt(2) + 1)) + 1/(sqrt(2) + 1)

 

Multiply sqrt(2) - 1 and 1 + sqrt(2) together using FOIL.

(sqrt(2) - 1) (sqrt(2) + 1) = -1 - sqrt(2) + sqrt(2)×1 + sqrt(2) sqrt(2) = -1 - sqrt(2) + sqrt(2) + 2 = 1:

(sqrt(2) + 1)/1 + 1/(sqrt(2) + 1)

 

Look for one in the denominator of sqrt(2) + 1.

(sqrt(2) + 1)/1 = sqrt(2) + 1:

sqrt(2) + 1 + 1/(sqrt(2) + 1)

 

Multiply numerator and denominator of 1/(sqrt(2) + 1) by the conjugate of the denominator.

Multiply numerator and denominator of 1/(sqrt(2) + 1) by sqrt(2) - 1:

1 + sqrt(2) + (sqrt(2) - 1)/((sqrt(2) + 1) (sqrt(2) - 1))

 

Multiply 1 + sqrt(2) and sqrt(2) - 1 together using FOIL.

(sqrt(2) + 1) (sqrt(2) - 1) = -1 + 1 sqrt(2) - sqrt(2) + sqrt(2) sqrt(2) = -1 + sqrt(2) - sqrt(2) + 2 = 1:

1 + sqrt(2) + (sqrt(2) - 1)/1

Look for one in the denominator of sqrt(2) - 1.

(sqrt(2) - 1)/1 = sqrt(2) - 1:

1 + sqrt(2) + sqrt(2) - 1

 

Evaluate 1 + sqrt(2) - 1 + sqrt(2).

1 + sqrt(2) - 1 + sqrt(2) = 2 sqrt(2):

2 sqrt(2)

Guest Feb 2, 2018
 #2
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0

1.

 

Simplify the following:

6/(2 sqrt(3) - 3)

 

Multiply numerator and denominator of 6/(2 sqrt(3) - 3) by the conjugate of the denominator.

Multiply numerator and denominator of 6/(2 sqrt(3) - 3) by 2 sqrt(3) + 3:

(6 (2 sqrt(3) + 3))/((2 sqrt(3) - 3) (2 sqrt(3) + 3))

 

Multiply 2 sqrt(3) - 3 and 3 + 2 sqrt(3) together using FOIL.

(2 sqrt(3) - 3) (2 sqrt(3) + 3) = -3×3 - 3×2 sqrt(3) + 2 sqrt(3)×3 + 2 sqrt(3)×2 sqrt(3) = -9 - 6 sqrt(3) + 6 sqrt(3) + 12 = 3:

(6 (2 sqrt(3) + 3))/3

 

In (6 (2 sqrt(3) + 3))/3, divide 6 in the numerator by 3 in the denominator.

6/3 = (3×2)/3 = 2:

2 (2 sqrt(3) + 3)

Guest Feb 2, 2018

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