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!
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)
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)
