Squares and Perfect Squares
More Lessons for Arithmetic, Math Worksheets
What is the square of a number?
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”.
What are 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
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.
How to determine if a number is a perfect square?We can use prime factorization to check if a number is a perfect square.
How to find smallest positive whole number that is a perfect square and a multiple of a specific whole number? Examples:
1. What is the smallest positive whole number that is a perfect square and is a multiple of 24?
2. What is the smallest positive whole number that is a perfect square and is a multiple of 150?
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 :
Look for parentheses to group negative numbers that are to be squared.
Example:
Simplify (-3)2 and -32
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