Lesson 2: Powers and Exponents

Property

For any expression $a^n$, $a$ is a factor multiplied by itself $n$ times if $n$ is a positive integer.
The expression $a^n$ is read $a$ to the $n^{th}$ power. In $a^n$, the $a$ is called the base and the $n$ is called the exponent.
$a^n = a \cdot a \cdot a \cdot ... \cdot a$ ($n$ factors)
Special names are used for powers of 2 and 3:
$a^2$ is read as '$a$ squared'
$a^3$ is read as '$a$ cubed'

Examples

• The expression $5 \cdot 5 \cdot 5 \cdot 5$ can be written in exponential notation as $5^4$, where 5 is the base and 4 is the exponent.
• To write $y^3$ in expanded form, you write out the base $y$ multiplied by itself 3 times: $y \cdot y \cdot y$.
• To simplify $2^5$, you calculate $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$, which equals $32$.

Explanation

Exponents are a shortcut for writing repeated multiplication. The small exponent number tells you how many times to multiply the larger base number by itself. This makes it much faster to write out long calculations involving the same factor.

Property

A perfect square is a positive integer that can be expressed as $n^2$ where $n$ is a positive integer.
In other words, a perfect square is the result of multiplying a whole number by itself.

Examples

Property

If the side length of a square is $s$ units, then its area is $s^2$ square units.
This demonstrates why we call $s^2$ a "perfect square" - it represents the area of an actual square.
The area of a square with side length $s$ is given by the formula:

Examples