What is Meant by the Square Root of a Number?
The square root of a number is a number that when multiplied by itself, gives the original number. Every number has two square roots, positive and negative square roots. A radical sign ‘ \( \sqrt{~~} \) ’ is used to represent a square root and the number under the radical sign is known as the radicand. A perfect square is a positive number that is derived by multiplying the number by itself. An imperfect square is a number that cannot be reduced to a whole number as its root.
Square Root of 1 to 50
Here is a list of square roots from \( 1 \) to \( 50 \).
How do we Calculate the Square Root of a Number?
The square root of any number can be calculated in two different ways.
- Prime Factorization
As the name suggests, prime factorization is the process of writing a number as the product of prime factors. Prime numbers are the numbers that have only two factors, that is \( 1 \) and the number itself. We use prime factorization as a method to find
the square roots of perfect squares.
Example: Find the value of \( \sqrt{49} \).
The prime factorization of \( 49 \) gives \( 7 \times 7 \), which implies that,
\( 49 = 7 \times 7 \)
\( \Rightarrow 49 = 7^2 \)
\( \Rightarrow \sqrt{49} = \pm 7 \)
So, the square roots of \( 49 \) are \( 7 \) and \( -7 \).
- Long Division Method
The square roots of imperfect square numbers can be determined using the long division method. This long division method will be introduced in higher grades.
Solved Examples
Example 1: Find the value of \( 15\sqrt{45}-11\sqrt{32} \).
Solution:
\( 15\sqrt{45}-11\sqrt{32} \) Write the expression
\( \Rightarrow 15 \times 6.708 ~-~ 11 \times 5.656 \) Substitute \( \sqrt{45} = 6.708 \) and \( \sqrt{32} = 5.656 \)
\( \Rightarrow 100.62 ~-~ 62.216 \) Multiply
\( \Rightarrow 38.404 \) Subtract
So, \( 15\sqrt{45}-11\sqrt{32} = 38.404 \)
Example 2: What is the value of the equation \( 7\sqrt{27}+13\sqrt{25}-12\sqrt{47} \) ?
Solution:
\( 7\sqrt{27}+13\sqrt{25}-12\sqrt{47} \) Write the expression
\( \Rightarrow 7 \times 5.196 + 13 \times 5 ~-~ 12 \times 6.856 \) Substitute \( \sqrt{45} = 6.708 \) and \( \sqrt{32} = 5.656 \)
\( \Rightarrow 36.372 + 65 ~-~ 82.272 \) Multiply
\( \Rightarrow 19.1 \) Simplify
So, \( 7\sqrt{27}+13\sqrt{25}-12\sqrt{47} = 19.1 \)
Example 3: Alex wants to build a fence around a square shaped garden that has an area of \( 50 \) square feet. Help Alex find the length of each side so that he doesn’t have to spend time measuring each side in the traditional way.
Solution:
The area of the garden, \( A = 50 \) ft\( ^2 \)
Use the formula for the area of the square to find the length of each side.
\( A = a^2 \) Write the formula
\( 50 = a^2 \) Substitute the values
\( \sqrt{50} = a \) Apply square root on both sides
\( 7.071 = a \) Simplify
So, the length of each side of the garden is \( 7.071 \) feet.
From the square root of 1 to 50 chart we find that the value of root 43 is 6.55.
Yes, 49 is a perfect square number because the square root of 49 is 7 and -7, which are both integers.
The perfect squares from 1 to 50 are 1, 4, 9, 16, 25, 36 and 49.
The square root of a number is calculated by the following methods:
- Prime factorization
- Long division
As we know, the square root of odd numbers is odd, and the root of even numbers is even. Here 25 is an odd number, so its square root is an odd number.