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

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

Algebraically, we can represent square numbers as:

Exponent Form: \( 7^2\)

Expanded Form: \( 22^2=22 \cdot 22 \)

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.

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:

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.

The factor tree shows we can write \( 1600 \) as:

\( 1600 = 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 2 \cdot 2 \cdot 2 \)

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

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.