Exponents (Negative Exponents)

Exponents (sometimes called powers) are used when you want to multiply a number (or variable) times itself a certain number of times.  Technically, because anything raised to the power of 0 is one (e.g., $8^0=1$), when you raise a number to a higher power, you are multiplying 1 by the base number as many times as the exponent tells you to.

$$\eqalign{8^0&=1\\8^1&=1 \times 8 = 8\\8^2&=1 \times8\times8=64\\8^3&=1\times8\times8\times8=512}$$

You see the pattern.

So, what are negative exponents?  Think about it: negative is the opposite of positive.  What is the opposite of multiplying?  Dividing!

Negative exponents tell you how many times to divide 1 by the base number.

$$\eqalign{8^0&=1\\8^{-1}&=1 \div 8 = \dfrac{1}{8}\\8^{-2}&=1 \div8\div8=\dfrac{1}{8\times8}=\dfrac{1}{64}\\8^{-3}&=1\div8\div8\div8=\dfrac{1}{8\times8\times8}=\dfrac{1}{512}}$$

Do you see that pattern?

Essentially, when you see a negative exponent, you calculate the exponent and put it underneath 1 in a fraction.

An assortment:

$$\eqalign{2^{-2}&=\dfrac{1}{2^2}=\dfrac{1}{4}\\4^{-1}&= \dfrac{1}{4^1}=\dfrac{1}{4}\\4^{-3}&=\dfrac{1}{4^3} = \dfrac{1}{64}}$$

NOTE: Negative exponents turn numbers into fractions but they do not make numbers negative.  The sign of the exponent DOES NOT affect the sign of the base.

It's interesting (but not always obvious) that when a number with a negative exponent is on the bottom of a fraction, then it flips to the top!

That means that:

$\dfrac{1}{8^{-2}} = \dfrac{8^2}{1} = \dfrac{64}{1}={64}$

Why does that work?  it makes sense if you think about it:

$$\eqalign{\dfrac {1}{8^{-2}}\\&= 1 \div (1\div 8\div8)&&\text{The denominator is 1 divided by 8 two times}\\&= 1 \div (1 \div64)&&\text{Simplify}\\&=\dfrac{1}{1}\div\dfrac{1}{64}&&\text{Rewrite as a division problem with fractions}\\&=\dfrac{1}{1}\times\dfrac{64}{1}&&\text{Remember when you divide with fractions you multiply by the reciprocal}\\&=\dfrac{64}{1}\\&=64}$$

It also means that:

$\dfrac{4}{2^{-3}} = \dfrac{4\times8}{1}={32}$

Here's how that works. 

$$\eqalign{\dfrac{4}{2^{-3}} &= \dfrac{4}{2^{-3}}\Lsh && \text{Negative exponents move bases to the other side of the fraction}\\ &=\dfrac{4 \times 2^3}{1} &&\text{Exponents drop the negative sign as they flip in the fraction}\\&=\dfrac{4 \times 8}{1}&&2^3=2\times2\times2=8\\&=\dfrac{32}{1}\\&=32 &&\text{Remember, any fraction over 1 is just a whole number}}$$

Finally, it also means that:

$\dfrac{5^{-2}}{5^{-3}} = \dfrac{5^{3}}{5^{2}}=\dfrac{1}{5}$

Here's how that works. 

$$\eqalign{\dfrac{5^{-2}}{5^{-3}} &= \dfrac{5^{3}}{5^{2}}\updownarrow && \text{Negative exponents move bases to the other side of the fraction and negative signs drop}\\&=\dfrac{125}{25}&&5^{3}=5 \times 5 \times 5=125 \text{ and } 5^{2}=5\times 5=25\\&=\dfrac{5}{1} = 5}$$See the "Alternate Methods" section below for another way to do this kind of problem.