Topic A: Exponential notation and properties of integer exponents

Want to learn more about these properties? Check out this video and this video.

Product of powers

This property states that when multiplying two powers with the same base, we add the exponents.

$x^n\cdot x^m=x^{n+m}$

Example

$5^2\cdot 5^5=5^{2+5}=5^7$

[[☃ explanation 1]]

Practice

[[☃ graded-group-set 1]]

Want to try more problems like these? Check out this exercise.

Quotient of powers

This property states that when dividing two powers with the same base, we subtract the exponents.

$\dfrac{x^n}{x^m}=x^{n-m}$

Example

$\dfrac{3^8}{3^2}=3^{8-2}=3^6$

[[☃ explanation 2]]

Practice

[[☃ graded-group-set 2]]

Want to try more problems like these? Check out this exercise.

Power of a power property

This property states that to find a power of a power we multiply the exponents.

$\left(x^n\right)^m=x^{n\cdot m}$

Example

$\left(8^2\right)^3=8^{2\cdot3}=8^6$

[[☃ explanation 3]]

Practice

[[☃ graded-group-set 1]]

Want to try more problems like these? Check out this exercise.

Power of a product

This property states that when taking the power of a product, we find the power of each factor and multiply.

$(x\cdot y)^n=x^n\cdot y^n$

Example

$(3\cdot 5)^6=3^6\cdot 5^6$

[[☃ explanation 3]]

Practice

[[☃ graded-group-set 3]]

Want to try more problems like these? Check out this exercise.

Power of a quotient

This property states that when taking the power of a quotient, we find the power of the numerator and the power of the denominator and divide.

$\left(\dfrac xy\right)^n=\dfrac{x^n}{y^n}$

Example

$\left(\dfrac72\right)^8=\dfrac{7^8}{2^8}$

[[☃ explanation 1]]

Practice

[[☃ graded-group-set 2]]

Want to try more problems like these? Check out this exercise.

Definition for negative exponents

We define a negative power as the multiplicative inverse of the base raised to the positive opposite of the power:

$x^{-n}=\dfrac{1}{x^n}$

Want to learn more about this definition? Check out this video.

Examples

• $3^{-5}=\dfrac{1}{3^5}$
• $\dfrac{1}{2^8}=2^{-8}$
• $y^{-2}=\dfrac{1}{y^{2}}$
• $\left(\dfrac{8}{6}\right)^{-3}=\left(\dfrac{6}{8}\right)^{3}$

Practice

[[☃ graded-group-set 2]]

Want to try more problems like these? Check out this exercise.

Some intuition

So why do we define negative exponents this way? Here are a couple of justifications:

Justification #1: Patterns

Notice how $2^n$ is divided by $2$ each time we reduce $n$. This pattern continues even when $n$ is zero or negative.

Justification #2: Exponent properties

Recall that $\dfrac{x^n}{x^m}=x^{n-m}$. So...

$\begin{align}
\dfrac{2^2}{2^3}&=2^{2-3}
\\
&=2^{-1}
\end{align}$

We also know that

$\begin{align}
\dfrac{2^2}{2^3}&=\dfrac{\cancel 2\cdot\cancel 2}{\cancel 2\cdot\cancel 2\cdot 2}
\\
&=\dfrac12
\end{align}$

And so we get $2^{-1}=\dfrac12$.

Also, recall that $x^n\cdot x^m=x^{n+m}$. So...

$\begin{align}
2^2\cdot 2^{-2}&=2^{2+(-2)}
\\
&=2^0
\\
&=1
\end{align}$

And indeed, according to the definition...

$\begin{align}
2^2\cdot 2^{-2}&=2^2\cdot\dfrac{1}{2^2}
\\
&=\dfrac{2^2}{2^2}
\\
&=1
\end{align}$