Question

Show that \[\frac{\sqrt{x^2 + x}}{x} = \sqrt{\frac{1}{x} + 1}\]
I'm not sure how to go about the algebraic manipulation to turn it into that result.

Answer 1

You see the x on the left hand side?

In order to bring that under the square root above, you need to square it (so that its square root will take you back to x again), then just divide through...

Answer 2

So then I get: \[\frac{\sqrt{\sqrt{x^2 + x}^2}}{\sqrt{x^2}}\] Is this right?

Answer 3

It's just sqrt(x^2 + x/x^2)

Answer 4

Yea, but doesn't squaring the terms change the value of the expression?

Answer 5

You start with x on the bottom and bring it under the square root as x^2

This is just a rearrangement, x is same as sqrt(x^2)

Answer 6

Oh, so I don't have to square then square root again for the numerator, only for the denominator

Answer 7

\[\frac{ \sqrt{x ^{2}+x} }{ \sqrt{x ^{2}} }\]

Answer 8

Right, you are only "moving" the x "into" the square root, so that is all you need to square

Answer 9

Then you can rewrite like\[\sqrt{\frac{ x ^{2}+x }{ x ^{2} }}\]Can you simplify from there?

Answer 10

Yea, if I knew I could do that it's easy but I didn't know you were allowed to do that in algebra

Answer 11

As, you said, the key thing is not to change the value, as long as you don't do that, no problem

Answer 12

its just manipulation, making the equation look different without really changing anything. Like @estudier said, you can change the x in the denominator to \[\sqrt{x ^{2}}\]because that is just x written differently

Answer 13

you can think of it in reverse, if I give you

sqrt(a^2/b^2) you can bring out the b^2 as b -> sqrt(a^2)/b