## Lesson 1: Representing Square Numbers

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:

#### Try these questions:

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

$$\textcolor{blue}{(-3)}^\textcolor{red}{2} = \textcolor{blue}{(-3)} \cdot \textcolor{blue}{(-3)} = 9$$

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:

$$\textcolor{blue}{(-7)}^\textcolor{red}{2} = \textcolor{blue}{(-7)} \cdot \textcolor{blue}{(-7)} = 49$$

$$\textcolor{blue}{-7}^\textcolor{red}{2} = -(\textcolor{blue}{(7)} \cdot \textcolor{blue}{(7)}) = -(49) = -49$$