Can someone please explain to me how this works?

[zeromodz]

If let's say we have the following expression

√(x^2 - y^2)

If I wanted to factor out x, then why can't I just take x out because a square root of a square is the base.

x√(1 - y^2)

But it turns out that the answer to this is incorrect and the answer to factoring out x is.

x√(1 - y^2/x^2)

Why is it I divide by y^2, by x^2 ? I have no idea! Thanks.

 

[NobodySpecial]

Suppose it was (x-y)^2 you couldn't take out the x because it also effects the y,
ie. (x-y)^2 = (x-y)(x-y) = x^2 -2xy + y^2

and sqrt(x^2-y^2) is (x^2-y^2)^0.5 , same principle the x goes with the y

 

[symbolipoint]

Use basic principles about fractions and the meaning of Multiplicative Inverse. You want to find an equivalent expression to your original one but you wish to show a factor of x².

Look at the expression under the radical symbol.
$$x^2 - y^2$$

If you DIVIDE by x² then you must also multiply by the reciprocal of x² to state the same meaning of the expression.
$$x^2 (1 - \frac{y^2}{x^2})$$

With that you can then find the square root can be simplified with a factor of just x outside of the radical.

 

[zeromodz]

Suppose it was (x-y)^2 you couldn't take out the x because it also effects the y,
ie. (x-y)^2 = (x-y)(x-y) = x^2 -2xy + y^2

and sqrt(x^2-y^2) is (x^2-y^2)^0.5 , same principle the x goes with the y

I understand, but how does dividing y^2 by x^2 resolve the expression?

 

[zeromodz]

Use basic principles about fractions and the meaning of Multiplicative Inverse. You want to find an equivalent expression to your original one but you wish to show a factor of x².

Look at the expression under the radical symbol.
$$x^2 - y^2$$

If you DIVIDE by x² then you must also multiply by the reciprocal of x² to state the same meaning of the expression.
$$x^2 (1 - \frac{y^2}{x^2})$$

With that you can then find the square root can be simplified with a factor of just x outside of the radical.

Okay thank you so much, I understand now!