Hello

I have an equation $\displaystyle \frac {\sqrt{x}-2}{(\sqrt{x} -1)^2} = 0$

If I try to solve this by multiplying both sides by same as denominator I get into complicated scenarios. Is there an easy way to handle this sort of thing?

Angus

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- Oct 16th 2011, 08:13 AMangypangySetting equation = 0 with multiple fractional powers
Hello

I have an equation $\displaystyle \frac {\sqrt{x}-2}{(\sqrt{x} -1)^2} = 0$

If I try to solve this by multiplying both sides by same as denominator I get into complicated scenarios. Is there an easy way to handle this sort of thing?

Angus - Oct 16th 2011, 08:16 AMQuackyRe: Setting equation = 0 with multiple fractional powers
Do you get complicated scenarios? If I multiply both sides by the denominator, I get:

$\displaystyle \sqrt{x}-2=0$

But I wouldn't even have to do that. If a fraction is equal to 0, then the numerator has to be 0. There is no other possibility. So by glancing, I would immediately be able to come to the above conclusion. - Oct 16th 2011, 08:33 AMangypangyRe: Setting equation = 0 with multiple fractional powers
- Oct 16th 2011, 08:49 AMQuackyRe: Setting equation = 0 with multiple fractional powers
Basically it's because $\displaystyle 0$ divided by anything is $\displaystyle 0$.

Let's start with $\displaystyle 0$.

Do you understand that $\displaystyle 0=\frac{0}{2}$?

And that $\displaystyle \frac{0}{56}=0$?

So... $\displaystyle 0=\frac{0}{1}=\frac{0}{2}=\frac{0}{3}=\frac{0}{4}= ...$

If, as a completely random example, $\displaystyle \frac{x+7}{3}=0...$

...and, of course, $\displaystyle 0=\frac{0}{3}$

Then $\displaystyle \frac{x+7}{3}=0=\frac{0}{3}$

Can you see here that this means that the numerators are equal?

With fractions, the only possible way you can get a result of 0 is if the numerator of the fraction is $\displaystyle 0$. - Oct 16th 2011, 09:00 AMangypangyRe: Setting equation = 0 with multiple fractional powers
Ah I see. So the denominator is irrelevent. It could be anything. [Presumably the only thing it can't be is zero - because that would result in an infinite number].

- Oct 16th 2011, 09:05 AMQuackyRe: Setting equation = 0 with multiple fractional powers
Absolutely (Nod)! Another way to think about it:

$\displaystyle \frac{x+7}{3}=0$

Multiply both sides by $\displaystyle 3$ (the denominator):

$\displaystyle \frac{3(x+7)}{3}=0\times~3$

$\displaystyle x+7=0$

Again, the denominator's value is lost.

So in your original problem, we just have to glance once at the seemingly difficult equation and find that it's actually just asking you to solve:

$\displaystyle \sqrt{x}-2=0$