An exponent is a short-hand notation to represent repeated multiplication of a base by itself. A number to the power of an exponent means to multiply that number by itself as many times as the exponent. It is a "repeated multiplication". This expression can be read as "3 to the power of 5" or "3 to the fifth power".
| \(\textcolor{blue}{3}^\textcolor{red}{5}=\) | \(\textcolor{blue}{3}\) | \(\cdot\) | \(\textcolor{blue}{3}\) | \(\cdot\) | \(\textcolor{blue}{3}\) | \(\cdot \) | \(\textcolor{blue}{3}\) | \(\cdot \) | \(\textcolor{blue}{3}\) |
| \(\uparrow\) | \(\uparrow\) | \(\uparrow\) | \(\uparrow\) | \(\uparrow\) | |||||
| \(\textcolor{red}{\scriptsize 1}\) | \(\textcolor{red}{\scriptsize 2}\) | \( \textcolor{red}{\scriptsize 3}\) | \(\textcolor{red}{\scriptsize 4}\) | \(\textcolor{red}{\scriptsize 5}\) |
Writing out an exponent is called expanded form. Condensing a long multiplication expression using exponents is called exponent form. Solving the expression is called standard form.
| Exponent Form | Expanded Form | Standard Form |
|---|---|---|
| \( \textcolor{blue}{15}^\textcolor{red}{4} \) | \(\textcolor{blue}{15} \cdot \textcolor{blue}{15} \cdot \textcolor{blue}{15} \cdot \textcolor{blue}{15}\) | \(50625\) |
A variable can also have an exponent.
\( \textcolor{blue}{a}^\textcolor{red}{4}= \textcolor{blue}{a} \cdot \textcolor{blue}{a} \cdot \textcolor{blue}{a} \cdot \textcolor{blue}{a} \)
\( \textcolor{blue}{b}^\textcolor{red}{2}= \textcolor{blue}{b} \cdot \textcolor{blue}{b} \)
Try changing the exponent and base:
A square number is the answer to an integer (not a fraction or decimal) with an exponent of \(2\). To square a number, simply multiply it by itself. A square number can be presented by a square, with equal length and width. For example, \(9\) is a square number, since you can draw out a square with \(9\) objects. You can express \(9\) as \(3\) with an exponent of \(2\).
\( \textcolor{blue}{3}^\textcolor{red}{2}= \textcolor{blue}{3} \cdot \textcolor{blue}{3} \)
This can be read as "3 to the power of 2", "3 to the second power", or "3 squared".
Examples of square numbers include: \( 25 \; (5^2), 81 \; (9^2), 100 \; (10^2) \) and \( 169 \; (13^2) \).
You can also square a negative number. Remember that multiplying two negative numbers together gives a "positive answer.
\( \textcolor{blue}{(-3)}^\textcolor{red}{2} = \textcolor{blue}{(-3)} \cdot \textcolor{blue}{(-3)} = 9 \)
Choose a number to square:
Remember order of operations! There is a difference between the two examples below:
| \( \textcolor{blue}{(-7)}^\textcolor{red}{2} \) | \( \textcolor{blue}{-7}^\textcolor{red}{2} \) |
According to BEDMAS, the exponent on the \( 7 \) occurs before the negative is applied like so: