Square Root Definition
The square root of a number is a factor of the number, which when multiplied by itself results in the number. Square roots are denoted by the symbol \(‘\sqrt{ }’\) called the radical.
Let’s assume a number x multiplied by itself results in a number y, then we can say that y is a square of x, or x is a square root of y.
Mathematically this is written as \(\sqrt{y} = x\) and read as the square root of y is x.
Square Root of 16
The number 16 is relatively small in value so we will calculate its square root by the prime factorization method:
Step 1: Find the prime factors of 16 using its factor tree.
\(16 = 2 \times 2 \times 2 \times 2\)
Step 2: Make pairs of identical factors.
\(16 = \left ( 2 \times 2 \right ) \times \left ( 2 \times 2 \right )\)
Step 3: Take a factor from each pair and multiply them to get the square root.
\(\sqrt{16} = 2 \times 2\)
\(\sqrt{16} = 4\)
So, the square root of 16 is 4.
Negative Square Root of 16
Each positive number has two square roots, positive and negative, and both the positive and negative numbers have the same absolute value.
Let us understand this with an example:
\(\left ( -\text{ }4 \right ) \times \left ( -\text{ }4 \right )= 16\) [Product of two negatives results in a positive]
Also, multiply 4 by itself,
\(4 \times 4 = 16\)
From this calculation, the square of -4 is 16, and the square of 4 is also 16, so we can say that the square roots of 16 are both 4 and -4.
Hence, \(\sqrt{16} = \pm\text{ }4\)
In general, \(\sqrt{y} = \pm\text{ }x.\)
Solved Examples
Example 1: A square has an area 16 square meters. Find the side length of the square.
Solution:
\(Area, A = side^{2}\) [Write the formula for area of square]
\(16 = side^{2}\) [Substitute 16 for A]
\(\sqrt{16} = \sqrt{side^{2}}\) [Take the square root of each side]
\(\pm 4 = side\) [Square root value]
We know that length cannot be negative.
So, the side length of the square is 4 inches.
Example 2: Evaluate \(\left( 3\sqrt{16} \times 1\right) – \left( \sqrt{16} \times 2\right).\)
Solution:
\(\left( 3\sqrt{16} \times 1\right) – \left( \sqrt{16} \times 2\right)\) [Write the equation]
\(=\left( 3 \times 4 \times 1 \right) – \left( 4 \times 2 \right)\) [Substitute 4 for \(\sqrt{16}\)]
= 12 – 8 [Multiply]
= 4 [Add]
So, \(\left( 3\sqrt{16} \times 1\right) – \left( \sqrt{16} \times 2 \right) = 4.\)
Example 3: The area of a circular park is 50.24 square feet. Calculate the radius of the ground.
Solution:
\(A = \pi r^{2}\) [Write the formula for area of circle]
\(50.24 = 3.14 \times r^{2}\) [Substitute 50.24 for A and 3.14 for \(\pi \)]
\(16 = r^{2} \) [Divide by 3.14 on each side]
\(\sqrt{16} = \sqrt{r^{2}}\) [Take positive square root of each side]
4 = r [Simplify]
So, the radius of the park is 4 feet.
Square root of a number is useful in solving area based problems on circles and squares.
The numbers whose square roots are integers are called perfect squares.
The number written under the square root symbol, ‘√’ is called the radicand. For example, in √16, 16 is the radicand under the radical symbol.