conditional evaluationa=(3-square root 2)^0.5 b=(3+square root 2)^0.5 b(1/a- b)=...?

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hala718's profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted on

a = ( 3- sqrt2)^1/2

b = ( 3+ sqrt2)^1/2

We need to find the values of  b(1/a - b)

First we will rewrite:

b*(1/a - b) = b/a - b^2

Now we will calculate each terms.

==> b/a = (3+sqrt2)^1/2 / (3-sqrt2)^1/2

              = [(3+sqrt2)/(3-sqrt2)]^1/2

              = [( 3+sqrt2)^2 / (9 - 2)]^1/2

               = (3+sqrt2) /sqrt7

               = sqrt7*(3+sqrt2)/ 7

==> b/a = sqrt7*(3+sqrt2)/7 ...............(1)

Now we will calculate b^2

b^2 = [(3+sqrt2)^1/2]^2 = (3+sqrt2)..............(2)

==> b(1/a - b) = (sqrt7*(3+sqrt2)/7  - (3+sqrt2)

We will factor (3+sqrt2)

==> b(1/a-b) = (3+sqrt2) [ sqrt7/7 - 1)

                     =(3+sqrt2) (sqrt7 - 7) / 7

==> b(1/a - b) = (sqrt7 - 7) (3+sqrt2) / 7

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll re-write a = sqrt(3-sqrt2) and b = sqrt(3+sqrt2)

To evaluate b(1/a- b), we'll remove the brackets:

b(1/a- b) = b/a - b^2

b/a = sqrt(3+sqrt2)/sqrt(3-sqrt2)

b/a = sqrt(3+sqrt2)*sqrt(3-sqrt2)/(3-sqrt2)

b/a = sqrt(3+sqrt2)*sqrt(3-sqrt2)*(3+sqrt2)/(3-sqrt2)*(3+sqrt2)

b/a = sqrt7*(3+sqrt2)/7

b^2 = [sqrt(3+sqrt2)]^2

b^2 = 3+sqrt2

b/a - b^2 = sqrt7*(3+sqrt2)/7 - 3-sqrt2

b/a - b^2 = (3sqrt7 + sqrt14 - 21 - 7sqrt2 )/7

b/a - b^2 = [3(sqrt7 - 7) + sqrt2(sqrt7 - 7)]/7

b/a - b^2 = (sqrt7 - 7)(3+sqrt2)/7

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