Square and square root
One mathematical shape that every student is familiar with is the square. As well as being a shape, squaring is also a procedure that can be applied to a number or algebraic letter.
When we square a number, we multiply it by itself. The term comes from the method of calculating the area of a square of known side-length.
The symbol we use to show that a number is to be squared is 2 â a superscript 2. So if we want to say â6 squaredâ we could write it as 62.
The square below has a side length of 4 m, what is its area?
To calculate the area of this square, we would multiply 4 m by 4 m which gives us 16 m2. The square of 4 is 16.
Example
What is 16 squared?
Solution
If you had a square of side length 16 m, what would be its area?
16 m Ă 16 m = 256 m2. So 16 squared is 256.
Question
What is ab squared?
As before, to square a number (or combination of letters) we must multiply it by itself.
ab Ă ab = (ab)2 or a2b2
The general rule when squaring numbers with indices is that you double the power.
(Z3)2 = Z6
(p5)2 = p10
Square root
If we know the area of a square and want to calculate its side-length, then we are trying to find its square root. This can seem more difficult to achieve but you should know the square numbers up to 122 (or 12 Ă 12). So you can calculate the square root of these. For any other number you will need a calculator.
Example
Find the square root of 144.
Solution
We know that 12 Ă 12 = 144, therefore the square root of 144 is 12.
Question
Find the square root of 81.
You should know that 9 Ă 9 = 81. Therefore the square root of 81 is 9.
The symbol for square root is \(\sqrt{}\)
Some square roots can only be done with trial and improvement, or far more simply through the use of a calculator. To find the square root of a number using a calculator, press the \(\sqrt{}\) button followed by the number. Check you can do this correctly by finding the solutions to \(\sqrt{144}\) and \(\sqrt{81}\)
Example
A square has an area of 640.09 cm2, what is the perimeter of the square?
Solution
The first step we must take to solve this problem is to find the length of the side of the square. Typing \(\sqrt{640.09}\) into a calculator gives the result 25.3
As the perimeter of a square is the side length multiplied by 4, we have 4 Ă 25.3 = 101.2 cm
The general rule when square-rooting numbers with indices is that you halve the power.
\(\sqrt{{t}^{6}}={{t}^{3}}\)
\(\sqrt{{m}^{5}}={{m}^{2.5}}\)