What is a Square Root?
As the fact has been established, the square root of a number is a number that when multiplied by itself equals the given number.
Every positive integer has two square roots in which one is positive and the other is negative with the same absolute value. A square root is represented by the symbol ‘ \( \sqrt{~~} \) ‘, called a radical sign. The number inside the radical sign is called the radicand.
Formula for the Square Root of Numbers
We can find the square root of any number using the formula,
\( \sqrt{x}=x^{\frac{1}{2}}=\sqrt{y \times y} = y \)
Here, \( x \) is a positive integer and y is the factor of \( x \).
So \( y \) is the square root of \( x \).
[Note: The number zero has only one square root, that is \( 0 \) itself.]
Rapid Recall
Square Root Formula
Solved Examples
Example 1: What is the square root of the number \( 256 \)?
Solution:
Given number \( = 256 \)
By using the square root formula, we can find its square root.
That is,
\( \sqrt{x}=x^{\frac{1}{2}} \)
\( \sqrt{256}=(16^2)^{\frac{1}{2}} \) [We can write \( 256 \) as square of \( 16 \)]
\( ~~~~~~~~~=(16)^{\frac{2}{2}} \)
\( ~~~~~~~~~= (16)^1 \)
\( ~~~~~~~~~= 16 \)
Therefore, the square root of \( 256 \) is \( 16 \).
Example 2: A square carpet has an area of \( 196 \) square meters. Sophie decided to decorate the carpet with fringe lace on its sides. How many meters of fringe lace should Sophie buy?
Solution:
The length of fringe lace required will be equal to the perimeter of the square carpet.
To determine the perimeter we need the side length of the carpet.
Let’s use the formula for the area of a square to find the side length:
Area of square carpet \( = \) side\( ^2 \)
\( 196 = \) side\( ^2 \) [Substitute \( 196 \) for area]
\( \sqrt{196} = \) side [Positive square root on both sides]
\( \sqrt{14\times 14} = \) side [\( 196 \) can be written as \( 14 \times 14 \)]
\( 14 = \) side
So the side length of the square carpet is \( 14 \) meters.
Perimeter of square carpet \( = 4~\times \) side
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 4\times 14 \)
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 56 \)
Therefore, Sophie is going to buy \( 56 \) meters of fringe lace to decorate the carpet.
Example 3: If \( \sqrt{x}=20 \), find the value of \( x \).
Solution:
As stated, \( \sqrt{x}=20 \).
The value of ‘\( x \)’ can be found by simply squaring the given equation.
\( \sqrt{x}=20 \)
\( (\sqrt{x})^2=(20)^2 \) [Squaring on both sides]
\( (x^{\frac{1}{2}})^2=(20)^2 \) [Formula for square root]
\( x = 400 \) [Square of \( 20 \)]
Hence, the value of \( x \) is \( 400 \).
Any positive number will have two square roots both of which have the same absolute value but opposite signs. So one square root is positive and the other is negative.
When a number is multiplied by itself the result obtained is called the square of the number. On the other hand, the square root of a number is a number that when multiplied by itself gives the original number.
Zero has only one square root which is zero itself.