Unit 6: Exponents and Exponential Functions

6.1: Exponents

Basic Exponents

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

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

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: $\left(\frac{3}{5}\right)^3=\frac{3^3}{5^3}$


Order of Operations


Multiplication Law

See separate page.


Division Law

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

This text will be automatically replaced.

It's not the point, but you can see this in the calculator: and .

(^)/(^)

^

x


Power Law

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

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

Quotient Law

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


Negative Exponents

Negative Exponents

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


Zero Exponents

Zero Exponents

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


Fractional Exponents


Principal Square Root


Radicals

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

Product Law for Radicals

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


Quotient Law for Radicals

Quotient Law for Radicals

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