What is Square Root 120 ?
Finding the square of a number and its square root are inverse operations. You must multiply the number by itself in order to find the square of the number.
For example: To find the square of a number 6, we will multiply the number by itself, that is,
6 × 6 = 36
Similarly, to find the square root of 36, we will do the inverse operation.
A square root is represented by this radical symbol,‘√’.
Since, \(6^2~=~36\)
\(\Rightarrow~\sqrt{36}~=~\sqrt{6^2}~=~6\)
The square root of 120 \(=~\sqrt{120}~=~2~\sqrt{30}\).
Deriving the Square Root with Prime Factorization
Steps to calculate the square root of 120 by prime factorization method:
Step 1: Find the prime factors of 120
∴ 120 = 2 × 2 × 2 × 3 × 5
Step 2: Make pairs of identical numbers.
\(120~=~(2~\times~2)~\times~2~\times~3~\times~5 \)
Step 3: Take a number from each pair and multiply to get square root.
\(\sqrt{120}~=~\sqrt{(2~\times~2)~\times~2~\times~3~\times~5} \)
\(\Rightarrow ~\sqrt{120}~=~2~\sqrt{30} \)
Since the value of \( ~\sqrt{30}~=~5.477 \) (approx.)
\(\Rightarrow ~\sqrt{120}~=~2~\times~5.477~=~10.954 \)
So, the square root of 120 is \(2~\sqrt{30} \) or 10.954.
Deriving Square Root by Long Division Method
Steps to find the square root of 120 using the long division method are given as follows:
Step 1: We will pair up the digits by placing a bar above them, starting from the right.
1\(\overline{20}\) . \(\overline{00}\) \(\overline{00}\) \(\overline{00}\)
Step 2: We now identify a number such that any number squared produces a result that is less than or equal to the first pair. In this case, the first pair only contains 1 number, or 1. 1 squared yields the result 1. The first pair is subtracted from the number, and the second pair is added as the divisor, making 20.
Step 3: In order to make the new divisor, when multiplied by each individual number in the quotient, give the product less than the dividend, we must first take the double of the quotient and add a digit with the divisor along with its placement in the quotient. The dividend is then reduced by the amount of the new divisor.
Step 4: To get the new dividend, we take the quotient’s double and add a digit with the new divisor’s placement in the quotient, so that when the new divisor is multiplied by the quotient’s individual number, the result is less than the dividend.
Step 5: In the step above, the difference is discovered. In order to produce a product that is less than the new divisor, the quotient’s double is once more taken and used as a divisor along with the inclusion of one more digit.
Step 6: The procedure is repeated exactly as in step 5. Thus, the division is displayed as follows:
Therefore, the value of the square root of 120, that is, \(\sqrt{120}\) is 10.954.
Definition of Negative Square Root
Each positive number has two square roots, one is positive and other is negative as their absolute value is the same.
Let us understand this with an example,
\(2~\sqrt{30}~\times~2~\sqrt{30}~=~120\)
\(-~2~\sqrt{30}~\times~-~2~\sqrt{30}~=~120\) [Since, \((-)~\times~(-)~=~(+)\)]
From the above, the square of \(-~2~\sqrt{30}\) is 120 and square of \(2~\sqrt{30}\) is also 120. So, we can say that square roots of 120 are \(2~\sqrt{30}\) and \(-~2~\sqrt{30}\) both.
\(\Rightarrow ~\sqrt{120}~=~\pm ~2~\sqrt{30}\)
In general,
Rapid Recall
Solved Square Root of 120 Examples
Example 1: The area of a circular field is \(120~\pi~ft^2\). Find the radius of the field.
Solution:
Let the radius of the circular field be ‘r’ feet.
The area of the field\(~=~\pi~r^2\)
\(\Rightarrow ~120~\pi~=~\pi~r^2\) [Given area = \(120~\pi~ft^2\).]
\(\Rightarrow ~120~=~r^2\) [Taking square root on both sides]
\(\Rightarrow ~\sqrt{120}~=~r\)
∴ \(10.954~=~r\)
Therefore, the radius of the circular field is 10.954 ft.
Example 2: Find the value of 8 multiplied by \(\sqrt{120}\).
Solution:
\(8~\times~\sqrt{120}\)
Since the value of \(\sqrt{120}~=~10.954\)
\(8~\times~\sqrt{120}~=~8~\times~10.954\) [Multiply]
\(=~87.632\)
Hence, the value of 8 multiplied by \(\sqrt{120}\) is 87.632.
Example 3: Find the value of \((3~\sqrt{120}~\times~\sqrt{30})~-~(\sqrt{120}~\times~2~\sqrt{30})\) .
Solution:
We have: \((3~\sqrt{120}~\times~\sqrt{30})~-~(\sqrt{120}~\times~2~\sqrt{30})\)
\((3~\sqrt{120}~\times~\sqrt{30})~-~(\sqrt{120}~\times~2~\sqrt{30})~=~(3~\times~2~\sqrt{30}~\times~\sqrt{30})~-~(2~\sqrt{30}~\times~2~\sqrt{30})\) [By PEMDAS rule]
\(=~\left ( 6~\times~\sqrt{30^2} \right )~-~\left ( 4~\times~\sqrt{30^2} \right )\) [Simplify]
\(=~(6~\times~30)~-~(4~\times~30)\)
\(=~180~-~120\)
\(=~60\)
Therefore, the value of given expression \((3~\sqrt{120}~\times~\sqrt{30})~-~(\sqrt{120}~\times~2~\sqrt{30})\) is 60.
120 has two square roots.
No, 120 is not a perfect square, as the square root of 120 is 10.954 which is in decimal form.
The square root of 120 is 10.9544511501.
So, the square root 120 is an irrational number because it is a non-terminating and non-repeating value.