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Square Root of 120 is the value that when multiplied by itself gives the number 120 as the product.
- The square root of 120 is 10.954 approximately.
- It is represented as √120, where √ is called the radical sign.
- The square root of 120 has a decimal value and not a whole number.
- It is expressed as √120 in radical form and 1201/2 or 1200.5 in exponential form.
The number 120 is not a perfect square as its square root is an irrational number. Its value is irrational because it does not have a finite value, i.e. it has infinite decimal places.
Read More: NCERT Solutions for Class 8 Mathematics Squares and Square Roots
Key Terms: Square Root of 120, Multiplication, Square Root, Irrational Number, Prime Factorization, Decimal, Long Division
What is Square Root of 120?
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Square Root of a number is the inverse mathematical operation of a square. It is the number that gives the original number as the product on multiplication with itself. The square root of 120 is expressed as √120. The value of the square root of 120 rounded off to three decimal places is 10.954.
Square Root of 120 = √120 = 10.9545
Simplification of Square Root of 120
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The value of the square root of 120 is 10.954451150013. We know that the square root of a number is the value that when multiplied by itself gives the number as the product.
If ‘b’ is a square of ‘a’, it can be denoted as
b = a²
Now, take the square root on both sides, and we get,
√b = √(a²)
The root un-square it or cancels the square, and we get,
√b = a
Similarly, by writing the prime factors of 120, we can find its square root value. We can write,
120 = 2 x 2 x 2 x 3 x 5 …..(i)
Here, 2, 3, and 5 are the prime factors of 120.
Taking the root of both sides by pairing numbers present underneath the radical sign. Therefore, (i) can be written as
√120 = √(2 x 2 x 2 x 3 x 5)
√120 = 2√(2 x 3 x 5) = 2√30 ...(ii)
It is the radical form of √120. To find the absolute value, we have to write the value of √30. The value of √30 is 5.477. Therefore,
√120 = 2 x 5.477 = 10.954
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Is Square Root of 120 Rational or Irrational?
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A number that can be expressed in a form of p/q is called a rational number otherwise it is an irrational number. Decimals having non-terminating repeated numbers after the decimal points are rational numbers otherwise it is an irrational number.
√120 = 10.954511501….
The decimal value of √120 is a non-terminating decimal with non-repeating numbers. Thus, 10.954511501 cannot be written in the p/q form. Therefore, √120 is an irrational number.
Read More: Perfect Cube of Numbers
How to Find Square Root of 120?
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Square Root of 120 can be calculated using the long division method as follows:
- Step 1: Starting from the right of 120, group the digits in a pair of two.
- Step 2: Now, find a number such that the square of that number gives a value equal to or less than the first pair. Here the first pair has only ‘1’. Thus, a square of 1 gives product 1. Now, 1 is subtracted from the first pair and the next pair i.e. 20 is added as the divisor.
- Step 3: Now, double the quotient and place a digit with a divisor along with its placement in the quotient such that a new divisor when multiplied by an individual number in the quotient gives a value less than the dividend, and subsequently subtracts it from a dividend.
- Step 4: Thus, the new dividend is obtained. For this, we double the quotient and place a digit with the divisor and place it in the quotient such that if we multiply the divisor with the individual number in the quotient, the product obtained is less than the dividend.
- Step 5: Again we take the double of the quotient and use it as a divisor along with the placement of one more digit in such a way, the same digit is mentioned in the quotient so that the product is less than a new divisor.
- Step 6: Repeat the process in the same way until the remainder is zero.
Since the value of the square root of 120 is irrational, the division will not end and the quotient will have an infinite number of decimal values.
Read More: Squares and Square Roots MCQs
Solved Examples on Square Root of 120
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Given below are a few solved examples on the square root of 120:
| Example 1: Calculate the square root of 120 using the radical form. Solution: The square root of 120 can be written as, √120 = √(2 x 2 x 2 x 3 x 5) Here note, 120 is not the perfect square. √120 = 2√(2 x 3 x 5) √120 = 2√30 We know the value of √30 is 5.47 √120 = 2 x 5.47 = 10.9545 Thus, the square root of 120 is 10.9545. Example 2: Evaluate the value of √7680. Solution: We can simplify the given value as √7680 = √(120 x 64) Using the Distribution Property, √7680 = (√120) x (√64) We know, √64 = √(8 x 8) = 8 √120 = 10.954 Thus, √7680 = 10.954 x 8 √7680 = 87.63 |
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Things to Remember
- Square Root of 120 is a number which when multiplied by itself, gives 120 as the product.
- Square Root of 512 is 10.954 on rounding off.
- In radical form, it is expressed as √120 and (512)⅓ or (512)0.33 in the exponential form.
- The value of the square root of 120 is infinite, thus, it is irrational in nature.
- 120 is not a perfect square as its square root is not a whole number.
Sample Questions
Ques. Find the smallest value by which 120 is multiplied so that product is a perfect square. (3 Marks)
Ans. To calculate the smallest value by which 120 is multiplied so that product should be a perfect square, we have to factorize 120 and find its factors.
On the prime factorization of 120, we get
120 = 2 x 2 x 2 x 3 x 5
Pairing the common factors we have,
120 = 2 x 2 x 2 x 3 x 5 ….(i)
From (i) we can see that only 2, 3, and 5 are left unpaired. Thus,
Required Number = 2 x 3 x 5 = 30.
Therefore, the smallest number required is 30.
Ques. Calculate the square root of 120 by the average method. (3 Marks)
Ans. To calculate the square root of 120 using the average method, find the perfect squares close to 120 i.e perfect square lesser than 120 is 100 and a greater perfect square is 121. The square root of 100 is 10 and 11 of 121.
Now divide 121 by 11 we get.
120/ 10 = 12
Now, take the average of 10 and 12
(10 ÷ 12)/ 2 = 22/ 2 = 11
Now, divide 120 by 11
120/ 11 = 10.909
To find the square root of 120, take an average of 11 and 10.909 we have,
(10.909 + 11)/ 2 = 21.909/ 2 = 10.9545
Thus, the square root of 120 is 10.9545.
Ques. Where does √-120 lie on the real number line? (2 Marks)
Ans. √-120 does not lie on the real number line. Thus it is not comparable with other real numbers such as integers on real number lines.
The square of any real number results in a non-negative real number. So, the square root of a negative real number is a complex number (non-real number). Therefore, the square root of -120 is not real and does not lie on a real number line.
Ques. Check if 120 is a perfect square or not. (2 Marks)
Ans. To check if 120 is a perfect square or not, we need the factors of 120. Factors of 120 are 2, 2, 2, 3, and 5.
Here the prime factors 2, 3, and 5 do not pair up. Hence, 120 is not the perfect square.
Ques. If the length of a rectangle is √(120)/2 m and the breadth is √480 m. Find the area of a rectangle. (3 Marks)
Ans. Given, Length (l) = √120/2
We know,
√120 = 10.954
- Length (l) = √(120)/2 = 10.954/2 = 5.47 m
- Breadth (b) = √(480) = √(4 x 120)/ = 2√(120) = 2 x 10.954 = 21.908 m
Area of rectangle = l x b
Area of rectangle = (5.47 x 21.908) = 119.84
Hence, the area of rectangle is 119.84 m2.
Ques. Is √120 an irrational number? (1 Mark)
Ans. Yes,√120 is an irrational number as its decimal expansion is non-terminating and non-repeating.
Ques. Arrange √120, 2√120/π, and π² in the increasing order of their values. (3 Marks)
Ans. Simplify all the values given
We know,
- First Value: √120 = 10.954 .... (i)
- Second Value: π = 3.14 = 3.14 x 3.14 = 9.860 .... (ii)
- Third Value: 2√120/π = (2 x 10.954)/3.14 = 6.98 ...(iii)
Hence the order of increasing value is: 2√120/π < π² <√120
Ques. If x = √120 + 1/√1080 then find the value of x/2. (3 Marks)
Ans. Given,
x = √120 + 1/√1080
Simplifying the value we get,
x = √120 + [1/√(9 x 120)]
x = √120 + [1/(3√120)]
x = [3 (√120 x √120) + 1]/√120
x = [3(120) + 1]/√120
x = 361/√120 = 361/10.954
x = 32.955
So,
x/2 = 32.955/2 = 16.47
Hence, the value of x/2 is 16.47.
Ques. Is the Square Root of 120 infinite? (1 Mark)
Ans. Yes, the square root of 120 is infinite. The decimal expansion of √120 is infinite as it is non-terminating and non-repeating which means it has infinite decimal places on division.
Ques. Find the value of six times the square root of 120. (2 Marks)
Ans. According to the question,
6√120 =?
The value of the square root of 120 is 10.954. Thus,
6 x 10.954 = 65.724.
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