Square Root of 5
Finding the square root of a number is the converse of finding the square of a number.
The square root of 5 is written as \(\sqrt{5}\), with the radical sign ‘\(\sqrt{~}\) ‘ and the radicand as 5. The square root of 5 has a value that is approximately equal to 2.2360. This value is also what is known as non-terminating and non-repeating.
The precise value of \(\sqrt{5}\) is obtained by the long division method.
Perfect Squares and Imperfect Squares
The square root of a perfect square number is always an integer, whether positive or negative. In contrast, the value of the square root of imperfect squares is not an integer, and this value is a decimal value, which is non-terminating and non-repeating.
For example, 2 is the square root of 4, which is a perfect square, and 2.236, is the approximate value for the square root of 5, which is an imperfect square.
Note: The table lists the approximate value of the square roots of some numbers that can be memorized to determine the square roots of higher imperfect square numbers.
Number | Square root |
|---|---|
2 | 1.414 |
3 | 1.732 |
5 | 2.236 |
7 | 2.646 |
Deriving Square Root by the Long Division Method
Steps to find the square root of 5 using the long division method are given as follows:
Step 1: Write the number as shown below by adding the bar on the top of the numbers.
\(\bar{5}\) . \(\overline{00} \) \(\overline{00}\) \(\overline{00}\)
Step 2: Take a number whose square is less than or equal to 5.
So, \(2^2~=~4\), which is less than 5.
Step 3: Write the number 2 as the divisor and the quotient. Write 4 below 5 and subtract.
Here, Quotient = 2 and Remainder = 1.
Step 4: Bring down 00 and write it after 1, so the new dividend is 100. Add number 2 with the divisor, so we get 4 in the place of the divisor.
Step 5: Add a number right next to 4 to get a new divisor such that the product of a number with divisor is less than or equal to 100. Subtract the number from 100 to get the remainder.
Since 42 \(\times\) 2 = 84, which is less than 100.
Here, Quotient = 2.2 and Remainder = 16
Step 6: Repeat the previous two steps to obtain the quotient up to three decimal places.
Therefore, the value of the square root of 5, that is, \(\sqrt{5}\) is 2.236.
So, \(\sqrt{5}\) = 2.236 (up to three decimal places)
Note: The square root of 5 in the exponential format is written as \((5)^{\frac{1}{2}}\).
Rapid Recall
Solved Examples
Example 1: John made a square with used sheets of paper and the square had an area of \(500~in^2\). Find the side length of the square.
Solution:
Let the side of the square be ‘\(a\)’ inches.
The area of the square = \(a^2\)
\(500~=~a^2\) [Given area = \(500~in^2\)]
\(\sqrt{500}~=~\sqrt{a^2}\) [Taking square root on both sides]
\(\sqrt{5~\times~(2~\times~2)~\times~(5~\times~5)}~=~a\) [Prime factorization]
\(2~\times~5~\sqrt{5}~=~a\) [Simplify]
\(a~=~10~\times~2.236\) [Since, \(\sqrt{5}\) = 2.236]
∴ \(a~=~22.36~in\)
Hence, the side length of the square is 22.36 inches.
Example 2: Find the value of \(10~\times~\sqrt{5}\).
Solution:
\(10~\times~\sqrt{5}\)
Since the value of \(\sqrt{5}\) = 2.236
\(10~\times~\sqrt{5}~=~10~\times~2.236\) [Multiply]
⇒ 22.36
Hence the value of \(10~\times~\sqrt{5}\) is 22.36.
Example 3: Find the value of \((2\sqrt{5})~+~10~(\sqrt{5}~\times\sqrt{5})\) .
Solution:
Given expression: \((2\sqrt{5})~+~10~(\sqrt{5}~\times\sqrt{5})\)
\((2~\times~2.236)~+~10~\left ( \sqrt{5^2} \right )\) [\( \sqrt{5} \) = 2.236]
= \((4.472)~+~(10~\times~5)\) [Simplify]
= \(4.472~+~50\)
= \(54.472\)
Therefore, the value of given expression is \((2\sqrt{5})~+~10~(\sqrt{5}~\times\sqrt{5})\) is 54.472.
There are two square roots of 5 which are 2,236 and -2.236.
No, 5 is not a perfect square, as the square root of 5 is 2.236, which is in decimal form.
The square root of 5 is equal to 2.23606797749.
So, the square root 5 is an irrational number because it is a non-terminating and non-repeating value.
Long division method is used to find the square root of imperfect square numbers.