Squares and Perfect Squares

Squares

The square of a number means to multiply the number by itself.

Example :

The square of 3 is 3 × 3 = 9

The square of a number can be written in exponent notation such as 3² where 3 is base and 2 is the exponent.

3² is read as “three to the second power” or “three squared”.

Perfect Squares

Perfect squares are the squares of whole numbers.

Example :

1, 4, 9, 16, 25 and 36 are the first 6 perfect squares because

1² = 1 × 1 = 1
2² = 2 × 2 = 4
3² = 3 × 3 = 9
4² = 4 × 4 = 16
5² = 5 × 5 = 25
6² = 6 × 6 = 36

Checking For Perfect Squares

We use repeated division by prime factors to check whether a given number is a perfect square.

Example :

Check whether 441 is a perfect square

Solution:

441 = 3 × 3 × 7 × 7
= 3 × 7 × 3 × 7
= 21 × 21
= 21²

So, 441 is a perfect square.

This following video shows how to find smallest positive whole number that is a perfect square and a multiple of a specific whole number.

Square of negative numbers, decimals and fractions

We can also have the square of negative numbers, decimals and fractions.

When calculating the square of a number, take note of the following:

1. The square of a number is always positive.

Example :

(5)² = (–5) × (–5) = 25

(–7)² = (–7) × (–7) = 49

Observe two important properties of a square in the last example above:

a) The square of a negative number becomes a positive number.

b) The square of a signed number is the same as the square of the unsigned number, i.e. ( - 7)² = 49 = 7².

2. The square of a decimal will have twice the number of decimal places as the original decimal.

Example :

(0.3)² = 0.3 × 0.3 = 0.09 ( 1 d.p. after squaring becomes 2 d.p.)

(0.03)² = 0.03 × 0.03 = 0.0009 ( 2 d.p. after squaring become 4 d.p.)

10.2² = 10.2 × 10.2 = 104.04 ( 1 d.p. after squaring becomes 2 d.p.)

3. To square a fraction, multiply the numerator by itself and do the same for the denominator.

Example:

Take note that if a positive fraction which is less than 1 is squared, the result is always smaller than the original fraction.

4. To square a mixed number, change it to an improper fraction before calculating the square.

Example :

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