Lesson 2: Powers and Exponents

Property

For any number $a$ and any positive integer $b$ we express the product $a \cdot a \cdot a \cdots a$, where there are $b$ factors of $a$, as $a^b$.
The superscripted number is the exponent and tells us how many times to multiply the base (the non-superscripted number) by itself.

Examples

• To evaluate $5^3$, you calculate $5 \cdot 5 \cdot 5$. This equals $25 \cdot 5$, which is $125$.
• The expression $m \cdot m \cdot m \cdot m$ can be written in exponential form as $m^4$. Here, $m$ is the base and $4$ is the exponent.
• The expression $3x^2$ means $3 \cdot x \cdot x$, while $(3x)^2$ means $(3x) \cdot (3x)$, which equals $9x^2$. The parentheses change what the base is.

Explanation

Exponents are a powerful shortcut for repeated multiplication. The base is the number you're multiplying, and the small, raised exponent tells you how many times to use the base as a factor. It's much faster than writing it all out!

Property

Powers have special names when read aloud:

• $a^2$ is read as "$a$ squared"
• $a^3$ is read as "$a$ cubed"
• $a^n$ (for $n \geq 4$) is read as "$a$ to the $n$th power"

Examples

Property

Evaluate expressions involving powers by substituting specific values for variables.
A power is an expression of the form $a^n$ where $a$ is the base and $n$ is the exponent.
To evaluate, replace the variable with its given value and calculate the result.

Examples