Exponents can present a way to express repeated multiplication. Later in this unit we will see that exponents do much more, but this is a good starting point. The thing we are multiplying is called the *base*, and the number of times we're multiplying is the *exponent* or *power*.

If the exponent is $2$, we often say "squared." If the exponent is $3$, we often say "cubed." Otherwise, it's just "to the ... power."

Definition: Exponents

$b^n=\underbrace{b \cdot b \cdots b \cdot b}_{n\text{ times}}$

- $b$ is the
*base* - $n$ is the
*exponent*or*power*

Example

$3 \cdot 3=\underbrace{3 \cdot 3}_\text{2 times}=3^2$

Here we are multiplying $3$ by itself $2$ times: $3^2$.

$3$ is the *base* and $2$ is the *exponent* or *power*.

We would say "three squared" or "three to the second power."

It's not the point, but you can work this out to a single number on the calculator: . |
3^2 9 x |

Exponents will hit the top and bottom of a fraction the way you'd expect.

Example

$\left(\frac{3}{5}\right)^3 =\underbrace{\frac{3}{5} \cdot \frac{3}{5} \cdot \frac{3}{5}}_\text{3 times}= \frac{3^3}{5^3}$

Here we are multiplying $\frac{3}{5}$ by itself $3$ times: $\left(\frac{3}{5}\right)^3$. The exponent applies to the numerator and the denominator of the fraction: $\left(\frac{3}{5}\right)^3=\frac{3^3}{5^3}$

See separate page.

Division Law

This is sometimes called the *Quotient of Powers* rule.

$\frac{b^n}{b^m} = b^{n-m}$

When we are dividing two expressions with the same base, we *subtract* the exponents. (We'll come back to this one in a second.)

Example

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It's not the point, but you can see this in the calculator: and . |
(^)/(^)
^
x |

Power Law

This is sometimes called the *Power of Powers* rule.

$\left(b^n\right)^m = b^{nm}$

When we are raising a power *to* a power, we *multiply* the exponents.

Product Law

$\left(ab\right)^n = a^n b^n$

This will *not* work for addition or subtraction. It is *not* true, for example, that $\left(a+b\right)^2 = a^2 + b^2$. You will get wrong answers if you do this.

Quotient Law

$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

Negative Exponents

$b^{-n}=\frac{1}{b^n}$ and $\frac{1}{b^{-n}}=b^n$

Zero Exponents

$b^0=1$, if $b \ne 0$

Radicals

$b^{\frac{1}{n}}=\sqrt[n]{b}$ and $b^{\frac{m}{n}}=\sqrt[n]{b^m}=\left(\sqrt[n]{b}\right)^m$

Product Law for Radicals

$\sqrt[n]{ab}=\sqrt[n]{a} \cdot \sqrt[n]{b}$

Quotient Law for Radicals

$\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$