In these lessons, we will learn what are perfect squares and how to calculate the square root of a perfect square.
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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 3^{2} where 3 is base and 2 is the exponent.
3^{2} is read as “three to the second power” or “three squared”.
Perfect squares are the squares of whole numbers.
Example:
1, 4, 9, 16, 25 and 36 are the first 6 perfect squares because
1^{2} = 1 × 1 = 1
2^{2} = 2 × 2 = 4
3^{2} = 3 × 3 = 9
4^{2} = 4 × 4 = 16
5^{2} = 5 × 5 = 25
6^{2} = 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
= 21^{2}
So, 441 is a perfect square.
We can use prime factorization to check if a number is a perfect square.
Examples:
We can also have the square of negative numbers, decimals and fractions.
When calculating the square of a number, take note of the following:
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 = 7^{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.)
10.2^{2} = 10.2 × 10.2 = 104.04 ( 1 d.p. after squaring becomes 2 d.p.)
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.
Look for parentheses to group negative numbers that are to be squared.
Example:
Simplify (−3)^{2} and −3^{2}
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