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# Exponent division question

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when you divide with a number with an exponent on the bottom, why does the exponent decrease and how do you know how much to decrease it by?

Aug 21, 2017
edited by Guest  Aug 21, 2017

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Maybe this example will help explain it.

$$\frac{x^5}{x^2}$$

Since     x5  =  x * x * x * x * x     and     x2  =  x * x     .......

$$\frac{x^5}{x^2}=\frac{x\,\cdot\, x\,\cdot\, x\,\cdot\,x\,\cdot\,x}{x\,\cdot\,x}$$

If we take a number, say  5, multiply it by  4, then divide it by  4, we get  5  again. That is...

$$\frac{5\,\cdot\,{\color{green}4}}{{\color{green}4}}\,=\,5 \\~\\ \frac{a\,\cdot\,{\color{green}b}}{{\color{green}b}}\,=\,a$$

So.....

$$\frac{x\,\cdot\, x\,\cdot\, x\,\cdot\,x\,\cdot\,{\color{green}x}}{x\,\cdot\,{\color{green}x}}=\frac{x\,\cdot\,x\,\cdot\,x\,\cdot\,x}{x}$$

We can reduce the fraction by  x  again.

$$\frac{x\,\cdot\,x\,\cdot\,x\,\cdot\,{\color{green}x}}{{\color{green}x}}\,=\,x\,\cdot\,x\,\cdot\,x$$             (When  x  ≠  0 .)

And....

$$x\,\cdot\,x\,\cdot\,x\,=\,x^3$$

So, from this we can see that....

$$\frac{x^5}{x^2}\,=\,x^{5-2} \\~\\ \frac{x^a}{x^b}\,=\,x^{a-b}$$

If this doesn't answer your question, can you think of an example problem that would answer it?

Anytime an exponent is confusing, it might help to write out all of the factors.

Aug 21, 2017
edited by hectictar  Aug 21, 2017