Can someone please explain to me how this works?

In summary, to factor out x from the expression √(x^2 - y^2), we must divide y^2 by x^2 and then multiply by the reciprocal of x^2 to maintain the same meaning of the expression. This allows us to simplify the square root and show a factor of x outside of the radical. This principle is similar to the one used when factoring expressions like (x-y)^2.
  • #1
zeromodz
246
0
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.
 
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  • #2
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
 
  • #3
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.
[tex]x^2 - y^2 [/tex]

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

With that you can then find the square root can be simplified with a factor of just x outside of the radical.
 
  • #4
NobodySpecial said:
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?
 
  • #5
symbolipoint said:
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.
[tex]x^2 - y^2 [/tex]

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

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!
 

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