## Lesson 2: Recognizing Square Numbers

A square number can be expressed as a base to an exponent of 2.

Graphically, we can represent a square number in different ways:

 3 3

Algebraically, we can represent square numbers as:

Exponent Form: $$7^2$$

Expanded Form: $$22^2=22 \cdot 22$$

Choose a number to square:

Examples of square numbers include: $$25 \; (5^2), 81 \; (9^2), 100 \; (10^2)$$ and $$169 \; (13^2)$$

Although we can write $$\frac{4}{9}$$ as:

$$(\frac{2}{3})^2 = \frac{2}{3} \cdot \frac{2}{3} = \frac{4}{9}$$

$$\frac{4}{9}$$ is not a square number because it is not an integer.

#### Review these lessons:

By breaking up the following multiplication statement, we can show that 400 is a square number:

$$\textcolor{green}{25} \cdot \textcolor{orange}{16} =(\textcolor{green}{5} \cdot \textcolor{green}{5}) \cdot (\textcolor{orange}{4} \cdot \textcolor{orange}{4}) = (\textcolor{green}{5} \cdot \textcolor{orange}{4}) \cdot (\textcolor{green}{5} \cdot \textcolor{orange}{4})$$

$$= 20 \cdot 20 = 20^2 = 400$$

In fact, the multiplication of two square numbers always gives a square number:

$$\textcolor{green}{a^2} \cdot \textcolor{orange}{b^2} =(\textcolor{green}{a} \cdot \textcolor{green}{a}) \cdot (\textcolor{orange}{b} \cdot \textcolor{orange}{b}) = (\textcolor{green}{a} \cdot \textcolor{orange}{b}) \cdot (\textcolor{green}{a} \cdot \textcolor{orange}{b}) = (\textcolor{green}{a} \cdot \textcolor{orange}{b})^2$$

We can use a factor tree to show that a number is a square number:

• 1255
• 5
• 245
• 5
• 49
• 7
• 7

The factor tree shows we can write $$1255$$ as:

$$1255 = 5 \cdot 5 \cdot 7 \cdot 7 = (5 \cdot 7) \cdot (5 \cdot 7) = (35) \cdot (35) = 35^2$$

Notice that we group the same numbers within the brackets. This ensure we have the same number in each bracket which will create a square number.

• 1600
• 8
• 4
• 2
• 2
• 2
• 200
• 25
• 5
• 5
• 8
• 4
• 2
• 2
• 2

The factor tree shows we can write $$1600$$ as:

$$1600 = 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 2 \cdot 2 \cdot 2$$

$$= (2 \cdot 2 \cdot 2 \cdot 5) \cdot (2 \cdot 2 \cdot 2 \cdot 5) = (40) \cdot (40) = 40^2$$

• 40
• 2
• 20
• 2
• 10
• 2
• 5

The factor tree shows we can write $$40$$ as:

$$40 = 2 \cdot 2 \cdot 2 \cdot 5$$

$$40 = (2 \cdot 2) \cdot (2 \cdot 5) = (4) \cdot (10)$$

$$40 = (2 \cdot 2 \cdot 2) \cdot (5) = (8) \cdot (5)$$

Notice that there is no way to group the numbers to make to equal brackets, this means 40 is not a square number.