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I've been doing the Khan Academy math courses to brush up on my math foundations before starting my CS/math degree in the fall semester.

I just don't fully understand negative exponents, I stumbled upon the following exponent simplification:

Equation screenshot

I understand how they got to a^-12/b^8 as it's simply exponent properties. I also understand that raising something to a negative exponent is the same as 1/a^12. But the part I don't understand is the last step. Shouldn't it be (1/a^12)/(b^8)? Why are the numerator and the denominator suddenly multiplied?

Imperator
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2 Answers2

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Note that

$$a^{-12}=a^{0-12}=\frac{a^0}{a^{12}}=\frac{1}{a^{12}}$$

tommik
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$$\frac{a^{-12}}{b^8} = a^{-12}\cdot \frac{1}{b^8} =\frac{1}{a^{12}}\cdot \frac{1}{b^8} =\frac{1}{a^{12}\cdot b^8}$$

If you divide by two things, one after another, then it is the same as dividing by their product.

$$(1/x)/y = 1/(xy)$$

For example divide $30$ by $2$ and then by $3$ and you get $(30/2)/3=15/3=5$ which is the same as $30/6$.

This is really no different to what happens with subtraction. If you subtract two things, one after the other, it is the same as if you subtract the sum of the two things:

$$a-b-c = a-(b+c)$$

Jaap Scherphuis
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  • Thanks Jaap, this is very helpful. It just didn't intuitively make sense to me but I understand better now! – Imperator Jul 03 '20 at 11:40