Introduction
The square root of a number is a value that, when multiplied by itself, gives us the original number. Hence, finding the square root is the converse of finding the square of a number.
If ‘a’ is a positive integer, the square root of ‘a’ is represented as ‘\( \sqrt{a}\)‘.
In math the square root is represented by the symbol ‘\( \sqrt{~}\)’. It’s also known as a radical symbol, and the number written with this sign is known as the radicand.
In this example, the radicand is 4 in \( \sqrt{4}\). The numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 from 1 to 100 are called perfect squares, while the remaining numbers are non-perfect squares, meaning their square roots will be irrational numbers.
Square root in the radical form is represented as \( \sqrt{a}\)
Square root in the exponential form is represented as \( (a)^\frac{1}{2}\)
Where ‘a’ is a positive integer.
Listing the Square Root from 1 to 100
The chart below contains a list of numbers from 1 to 100 and their square roots. Many mathematical problems can be solved with these square root values.
How do you Find the Square Root of a Number from 1 to 100?
There are two methods to calculate the square root of numbers between 1 to 100. Those are:
- Prime factorization method
- Long division method
The factorisation method or the long division method can be used to find the square root of any number. When a number is a perfect square, the square root can be easily determined by the prime factorization method. However, if the numbers are imperfect squares, in decimal form, or are too large, we use the long division method.
For example, finding the square root of 81. Using the prime factorisation method.
What is the value of \( \sqrt{81}\)?
Prime factorization of 81 is \( 9~\times~9\).
Prime factors are paired as (9,9).
Therefore, the square root \( \sqrt{81}\) is 9.
Also, for the square root of 7, we use the long division method.
So what is the value of \( \sqrt{7}\)?
Step 1: Rewrite the number as shown below.
\( \overline{7}\) . \( \overline{00}\) \( \overline{00}\) \( \overline{00}\)
Step 2: Take a number whose square is less than or equal to 7.
\( 2^2~=~4\), which is less than 7. So we will take 2.
Step 3: Write the number 2 as the divisor and 7 as the dividend. Now divide 7 by 2.
Step 4: Bring down 00 and write it after 3, so the new dividend is 300 and add the quotient 2 to the divisor, that is, 2 + 2 = 4.
Step 5: Add a digit right next to 4 to get a new divisor such that the product of a number with the new divisor is less than or equal to 300.
46 \( \times\) 6 = 276, which is less than 300.
Subtract the product 276 from 300 to get the remainder.
Step 6: Repeat the previous two steps to obtain a quotient that has three decimal places.
Solved Examples
Example 1: Find the value of P in equation \( P~\sqrt{36}~=~2.15\) ?
Solution:
\( P~\sqrt{36}~=~2.15\) Write the equation
\( P~\times~6~=~2.15\) Substituted \( \sqrt{36}\) as 6
\( \frac{P~\times~6}{6}~=~\frac{2.15}{6}\) Divide each side by 6
\(P~=~\)0.35833….
Hence the value of P is 0.358 up to three decimal places.
Example 2: A square shaped brick wall has an area of 49 square meters. Find the length of the side of this wall?
Solution: Let ‘l’ be the length of the side of a brick wall.
\(A~=~l^2\) Write formula for area of square
\(49~=~l^2\) Substitute 49 for \(l\)
\(\sqrt{49}~=~\sqrt{l^2}\) Apply square root both side
\(7~=~l\) Simplify
So, the length of the side of a brick wall is 7 meters.
Example 3: The area of a circular field is 188.4 square feet. Find the radius of the field.
Solution:
\(A~=~\pi~r^2\) Write the formula for area
\(188.4~=~3.14~\times~r^2\) Substitute 188.4 for A and 3.14 for \(\pi\)
\(\frac{188.4}{3.14}~=~\frac{3.14~\times~r^2}{3.14}\) Divide each side by 3.14
\(60~=~r^2\) Simplify
\(\sqrt{60}~=~\sqrt{r^2}\) Apply square root both side
\(7.746~=~r\) Substitute 7.746 for \(\sqrt{60}\)
So, the radius of the circular field is 7.746 feet.
There are two methods for calculating the value of square roots from 1 to 100 that are commonly used. We can use the prime factorization method for perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, and 100) and the long division method for non-perfect squares (2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 99).
Because the numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 are perfect squares, their square roots will be whole numbers, so we can write them in a form of a/b where b ≠ 0. As a result, the square roots of 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 are rational.
Non-perfect squares include the numbers 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, and 99. As a result, their square root will be an irrational number, which cannot be expressed as of a/b where b ≠ 0.