3/3-sqrt 3=A(3+sqrt3) Where A=??

Question

HOW IN THE WORLD DO YOU DO THIS PROBLEM???

Answer 1

A = 3 / [(3-sqrt 3)(3+sqrt3)] ### = 3 / (9 - 3) = 3 / 6 = 1/2.

The key insight is to note that

(3-sqrt 3)(3+sqrt3)

is a factorization of the difference of two squares.

(a^2 - b^2) = (a - b)(a + b).

In your problem, a = 3, and b = sqrt 3.

I hope this helps.

Live long and prosper.

### By the way, you should enclose a complicated denominator by parentheses, so that it is clear that the first term isn't simply 3/3 = 1. (That is how it would be evaluated according to the standard order of preference for mathematical operations.)

The use of parentheses makes it clear that in 3/(3-sqrt 3), the value of (3-sqrt 3) is to be evaluated first BEFORE dividing it into the 3.

Answer 2

3/(3 - sqrt 3) = A(3 + sqrt 3) multiply both sides by 3 - sqrt3

3 = A(3 - sqrt 3)(3 + sqrt 3)

3 = A (3^2 - sqrt 3^2)

3 = A (9 - 3)

3 = 6A

Divide by 6

A = 1/2

Answer 3

A=(1-sqrtt3)/(3+sqrt3)*(3-sqer3/3-sqrt3)
=(1-sqrt3)(3-sqrt3)/(9-3)
=(1-sqrt3)(3-sqrt3)/6

Answer 4

3/(3-sqrt 3)=A(3+sqrt3)

3/[(3-sqrt 3)(3+sqrt3)]=A

3/ [ (3)^2 - (SQRT(3))^2] =A

3/ (9 - 3 ) = A

3/6 = A

A = 1/2