√x + √x-7 = 21/√x-7 my teacher sucks and doesn't teach so can anyone show me how you get 16?

Despite your negative comments about your teacher...

sqrt(x) + sqrt(x - 7) = 21 / sqrt(x - 7)

multiply everything by sqrt(x - 7) to clear the fraction...

(with the domain restriction that x > 7 in order for this to be defined)

sqrt(x) sqrt(x -7) + (x - 7) = 21 <== now combine the square roots and collect like terms

sqrt[x(x -7)] = 28 - x

( again, with the domain restriction that x > 7 in order for the original equation to be defined)

square both sides again

x(x - 7) = (28 - x)^2

x^2 - 7x = 28^2 - 56x + x^2

49x = 28^2 = 784

x = 784 / 49 = 16

x = 16

check:

sqrt(16) + sqrt(16 -7) = 21 / sqrt(16 - 7)?

4 + 3 = 21 / 3

7 = 7 checks

x = 16

edit:

if you were trying to rationalize a fraction like 21 / (sqrt(x) - 7), then you would multiply by its conjugate (sqrt(x) + 7)

but even if the equation were sqrt(x) + sqrt(x) - 7 = 21 / (sqrt(x) - 7), you would still multiply by the denominator itself to clear the fraction

√x + √(x-7) = 21/√(x-7)

Multiply both sides by √(x-7)

√x√(x-7) + x - 7 = 21

√(x^2-7x)) = 28 - x

Square both sides of equation

(x^2-7x) = (28 - x)^2

Expand right side, and then solve for x

x^2 - 7x = 28^2 - 56x + x^2

-7x + 56x = 28^2

x = (28^2/49) = 16

On a side note, I think you are being too hard on your teacher or are viewing the situation in a biased light, I doubt your teacher has a very receptive class to her (from the sounds of it) and it would be in the benefit of you and your classmates to not be a dick to her.

"it is the square root isof x-7 not just x."

You would write that as √(x-7).

http://www.flickr.com/photos/dwread/8275026425/