Multiplying Exponents (same base)

When you get into algebra, you'll often multiply terms that already have exponents in them.

When you multiply exponents with the same base, you can bring those exponents together into one term  -- and there are some good tricks for doing it quickly and efficiently.

Let's start with a simple problem:

$$\eqalign{2^2 \times 2^2 &=&&\text{This problem multiplies two exponents with the same base: 2}\\(2\times2) \times (2\times 2)&= &&\text{If you write this all the way out, it's just 2 multiplied by itself 4 times}\\2\times 2 \times2 \times 2&=16&&\text{You can remove the parentheses because it doesn't matter what order you multiply in}\\2^4&=16&&\text{You can also simplify.  2 times itself 4 times is } 2^4}$$

Is there an easier way to get to this point? Yes, there is. But, if you forget the trick, you can always  just write out all of your exponents, add up the number of bases you're multiplying together, and there's your new exponent!

A rule that you can use when multiplying exponents with the same base is:

When you multiply exponents with the same base, keep the base and add the exponents.

$$\eqalign{2^2 \times 2^2 \\&=2^{2+2}\\&=2^4\\&=16}$$

This trick will work anytime you are multiplying exponents with the same bases:

$$\eqalign{5^3 \times 5^2 \\&=(5 \times 5  \times 5) \times(5 \times 5)\\&=5^5\\&=3125\\\text{OR}\\&=5^{3+2}\\&=5^5\\&=3125}$$

Because exponents work the same way, no matter what the base is, this also works for fractions, decimals, variables, and terms that include exponents:

$$\eqalign{(\dfrac{2}{5})^2 \times(\dfrac{2}{5})^3\\\text{}\\\text{You can write it out:} \\&=(\dfrac{2}{5}\times\dfrac{2}{5})\times(\dfrac{2}{5}\times\dfrac{2}{5}\times\dfrac{2}{5})\\&=(\dfrac{2}{5})^5\\&=\dfrac{32}{3125}\\\text{OR, use the trick:}\\&=(\dfrac{2}{5})^{2+3}\\&=(\dfrac{2}{5})^5\\&=\dfrac{32}{3125}\\\text{}\\\text{}\\\text{And:}\\\text{}\\\large{x^8 \times x^3} \\\text{}\\\text{You can write it out:}\\&=(x\times x\times x\times x\times x\times x\times x\times x)\times(x\times x\times x)\\&=x^{11}\\\text{OR, use the trick:}\\&=x^{8+3}\\&=x^{11}}$$

Note: Unless there are parentheses, exponents ONLY apply to the number or variable they are attached to.  So, while $(2x)^2=(2x)\times(2x)$, $2x^2=2\times x\times x$.  Likewise, when there are exponents on fractions, exponents only apply to both the numerator and the denominator when the fraction is in parentheses.  If the fraction is not in parentheses, then the exponent only applies to the number that the exponent is attached to.

This trick also works with negative exponents.  Just follow the rules of adding negative numbers:

$$\eqalign{x^8 \times x^{-3}\\ &=x^{8+(-3)}\\&=x^5\\\text{}\\\text{}\\\text{}\\x^{-2}\times x^{-5}\\&=x^{(-2)+(-5)}\\&=x^{-7}}$$

For more about how negative exponents work, see Exponents (Negative exponents).