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June 6th, 2014, 12:00 AM   #1
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x^-2 = 1/x^2

Hi
Can someone point me to a clear explanation of why x^-2 = 1/x^2

I know that the rule is that negative exponents move to the other side of the fraction and become positive exponents. But why?
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June 6th, 2014, 01:52 AM   #2
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Multiply both sides of the equation by$\displaystyle x^2$...

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June 6th, 2014, 05:40 AM   #3
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It's more or less a definition, although a natural one.

We have $x^{n} = x \cdot x^{n-1}$. Satisfy yourself that this is true by putting some positive integers in place of $n$. Now, dividing both sides by $x$ we get
$$\frac{x^n}{x} = x^{n-1}$$
From this we can work down:
\begin{align*}
\frac{x^2}{x} &= x^1 = x \\
\frac{x^1}{x} &= x^0 = 1 \\
\frac{x^0}{x} &= x^{-1} = \frac{1}{x} \\
\frac{x^{-1}}{x} &= x^{-2} = \frac{1}{x^2} \\
&\cdots
\end{align*}
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Last edited by skipjack; June 6th, 2014 at 06:32 AM.
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June 6th, 2014, 05:41 AM   #4
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I like to look at what happens to reducing powers of x.

if we write it out as:

$\displaystyle x^5 = x.x.x.x.x$
$\displaystyle x^4 = x.x.x.x$
$\displaystyle x^3 = x.x.x$
$\displaystyle x^2 = x.x$
$\displaystyle x^1 = x$

We can see that to reduce the power each time we divide by x. Hence why

$\displaystyle x^0 = 1$ as this is $\displaystyle x^1 / x = x/x = 1$

then we keep going

$\displaystyle x^{-1} = 1/x$
$\displaystyle x^{-2} = 1/x.x = 1/x^2$

Hope that is clear enough to follow

PS: Don't know why it wont show my negative powers properly still new to this stuff

Last edited by nmenumber1; June 6th, 2014 at 05:57 AM.
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June 6th, 2014, 05:41 AM   #5
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this keeps happening to me, finish posting and by them someones already there with a better and neater explanation lol good job mate
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June 6th, 2014, 05:51 AM   #6
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Quote:
Originally Posted by nmenumber1 View Post
Don't know why it wont show my negative powers properly still new to this stuff
Curly braces are your friend.

Code:
x^{-1}
$$x^{-1}$$
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June 6th, 2014, 05:56 AM   #7
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ah nice one thanks man
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