Every non-negative real number has a unique non-negative square root, called the principal square root. Which is denoted by √x, where √ is called the radical sign or radix and the number under the radical sign here x is called the radicand.For example, the principal square root of 9 is 3, denoted by √9 = 3, because 32 = 3 x 3 = 9 and 3 is non-negative. The square root of 120 is that number which when multiplied by itself gives the number 120.
Number 120 is not a perfect square hence the square root of 120 results in a decimal number not the whole number.
Square Root of 120 (\[\sqrt{120}\]) = 10.954
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The square root of the rational number, 120 gives a decimal number and not a whole number, because the number 120 doesn’t have proper factors. Every non-negative real number has a non-negative square root, which is called the principal square root. Therefore, the square root of 120 gives a result as a unique non-negative number. It is denoted as √120, where √ is called radical sign or radix and the number 120, whose square root has to be derived is called radicand
Calculating the square root of an imperfect square is a bit of a complex task. Following are the methods that can be used to find the square root of a number.
Prime factorization method
Repeated subtraction method
Long division method
Number line method
Average method
This method works if the number is a perfect square, but, if the number is not a perfect square prime factorization method and the repeated subtraction method will not work, we have to use other methods for finding the square roots.
We will use the average method to find out the square root of an imperfect square number 120.
Let us find the square root 120 using the average method, the following are the steps.
Step 1: Find out the two perfect square numbers which are very close to the given number on either side. For example the number ‘120’, the immediate perfect square lesser than 120 is ‘100’ and the immediate perfect square greater than 120 is ‘121’.
Step 2 : Note down the square roots of the perfect squares, here the square root of ‘100’ is ‘10’ and the square root of ‘121’ is ‘11’.
Step 3 : Square root of a given number lies between the square roots of numbers determined in step 2.Square root of ‘120’ is any number between 10 and 11.
Step 4: Divide the number whose square root is determined by any of the numbers obtained in Step 2.‘120’ can be divided either by ‘10’ or ‘11’.
Let us divide ‘120’ by ‘10’
\[\frac{120}{10}\] = 12
Step 5: Find the average of the quotient and divisor in Step 4.
The average of 10 and 12 is
Average = \[\frac{10 + 12}{2}\] = 11
Step 6: Now divide 120 by step 5 answer
\[\frac{120}{11}\] = 10.9090
Step 7: Now, average 11 and 10.9090 by adding them together and dividing the sum by two you get 10.954545
Check your work by multiplying your answer by itself. If 10.9545 is multiplied by 10.9545 we get 120.001.
Therefore \[\sqrt{120}\] = 10.9545
One of the methods to find the square root of 120 is by finding the factors of 120.
Factors of 120 are 2 x 2 x 2 x 3 x 5.
Factor tree for 120 is
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Steps to find out the square root by factors
Step 1: find out the factors of the number here
120 = 2 x 2 x 2 x 3 x 5
Step2 : Now taking root of the factors.
\[\sqrt{120}\] = \[\sqrt{2 \times 2 \times 2 \times 3\times 5}\]
Step 3: taking out root means taking out the pair of common factors as a single common number outside the radical sign. Here only 2 are in pairs so taking out 2 out of the radical sign. And single 2 3 and 5 does not have pair so they remain inside the radical sign
= 2 x \[\sqrt{2 \times 3 \times 5}\]
Step 4: Multiplying, we get
= 2 x\[\sqrt{30}\]
Step 5: finding the root value
= 2 x 5.477
Step 6 : simplify
=10.9544
The most precise and convenient method to find out the square root of imperfect square is by long division method. Here is the stepwise calculation by long division method.
10.954 | |
1 +1 | 1 20 -1 |
20 +0 | 0 20 00 |
209 + 9 | 2000 - 1881 |
2185 +5 | 11900 - 10925 |
21904 + 4 | 97500 - 87616 |
21908 | 9884 |
Technically, just the "check mark" part is used in the radical; the line across the top is called the "vinculum".
Square Root Day- The 4th of April 2016 is a Square Root Day, because the date looks like a square root 4/4/16
The next after that is the 5th of May 2025 (5/5/25)
\[\sqrt[Happy]{\text{Square root Day}}\]
Example 1: Find the square root of 12
Solution:
Factor of 12 are 2 x 2 x 3
Taking root
= \[\sqrt{2 \times 3 \times 3}\]
= \[2\sqrt{3}\]
= 2 x 1.7320
=3.4641
Example 2 :Find the square root of 75
Solution:
Factor of 75 = 5 x 5 x 3
Taking the root of
= \[\sqrt{5 \times 5 \times 3}\]
= \[5\sqrt{3}\]
= 5 x 1.7320
= 8.66
Find the square root of 86 by long division method.
Find the square root of 45 by factor method.
1. How to Find Square Root by Repeated Subtraction?
Answer : We know that by the property of square numbers, if a natural number is a square number, then it has to be the sum of successive odd numbers starting from 1. We will be using this property to find out the square root of a number by repeated subtraction method.Therefore, you can find the square root of a number by repeatedly subtracting successive odd numbers from the given square number, till you get zero.
For Example : how to find the square root of 81 by repeated subtraction method.
81 - 1 = 80
80 - 3 = 77
77 - 5 = 72
72 - 7 = 65
65 - 9 = 56
56 - 11 = 45
45 - 13 = 32
32 - 15 = 17
17 - 17 = 0
Here we got the result 0 in the 9th step, so the square root of 81 is 9 i.e √81 = 9.
2. What is the Opposite of Square Root?
Answer: The square root of a number, is that number which when multiplied by itself gives the number itself. That number is called a perfect square. We calculate the area of a square as a product of side i.e side^{2}, but if we have to calculate the side of a square we need to take the square root of the area. Thus we can say that the square root is the inverse operation of squaring a number. Hence square of a number is the opposite of square root of a number.
For Example : square root of 100 is 10 while the square of 10 is 100.
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