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Square Root of 153
The square root of 153 is expressed as √153 in the radical form and as (153)^{½} or (153)^{0.5} in the exponent form. The square root of 153 rounded up to 9 decimal places is 12.369316877. It is the positive solution of the equation x^{2} = 153. We can express the square root of 153 in its lowest radical form as 3 √17.
 Square Root of 153: 12.36931687685298
 Square Root of 153 in exponential form: (153)^{½} or (153)^{0.5}
 Square Root of 153 in radical form: √153 or 3 √17
What Is the Square Root of 153?
The square root of a number is the inverse process of squaring a number.
 A square of a number y is x^{2 }⇒ y = x × x. 153 = 12.369 × 12.369
 √153 = √(12.369 × 12.369)
 153^{½} = (12.369^{2})^{½ }
 12.369 = 153^{½} ⇒ √153 = ±12.369
Is Square Root of 153 Rational or Irrational?
√153 = 12.3693168769. √153 cannot be written in the form of p/q, hence it is an irrational number. The value of the square root of 153 is a decimal number whose digits after the point is neverending or nonterminating.
How to Find the Square Root of 153?
The square root of 153 or any number can be calculated in many ways. Two of them are the approximation method and the long division method.
Square Root of 153 by Approximation Method
 Take two perfect square numbers which are just smaller than 153 and just greater than 153. √144 < √153 < √169
 12 < √153 < 13
 Using the average method, divide 153 by 12 or 13
 Let us divide by 12 ⇒ 153 ÷ 12 = 12.75
 Find the average of 12,75 and 12.
 (12.75 + 12) / 2 = 24.75 ÷ 2 = 12.375
 √153 ≈ 12.369
Square Root of 153 by the Long Division Method
The long division method helps us to find a more accurate value of square root of any number. Let's see how to find the square root of 153 by the long division method.
 Start pairing the digits from the right side of the number 153 by putting a bar on top of them. Here we will have two pairs: 53 and 1 (pairing from right).
 Now, find a number(a) whose square is ≤ 1. The value of a will be 1 as 1 × 1 = 1≤ 1.
 We get the quotient (1). Now, we will add the divisor a with itself and get the new divisor.
 Drag the next pair down (new dividend becomes 053) and find a number (A) such that 2A × A ≤ 53. In this case, the value of A will be 2.
 Now, put a decimal in the dividend (153) after 3 and quotient (12) after 2 simultaneously. Also, add 3 pairs of zero in the dividend after the decimal (153. 00 00 00) and repeat the above step for the remaining three pairs of zeros.
So, we get the value of the square root of √153 = 12.369 by the long division method.
Explore square roots using illustrations and interactive examples
 Square root of 144
 Square root of 54
 Square root of 53
 Square root of 123
 Square root of 120
 Square Root of 15
Tips and Tricks
 The square root of 153 lies between the square root of 144 and 169. Therefore, 12 < √153 < 13.
 Use the average method then, to evaluate the approximate value of √153.
Important Notes
 √153 is 12.369 approximated to 3 decimal places.
 The simplified form of √153 in its radical form is 3√17.
 √153 is an irrational number.
Square Root of 153 Solved Examples

Example 1: Amy stacks 153 books in such a way that there are as many books in each row as the number of rows. How many books will she take out of the arrangement or how many books will she add to make such an arrangement?
Solution: She needs to find the least number to be added to 153 or subtracted from 153 to make it a perfect square. 153 lies between 144 and 169, which are the perfect squares close to 153.
153  9 = 144 ⇒ 12 × 12 = 144
153 + 16 = 169 ⇒ 13 × 13 = 169
If Amy decides to have 12 books arranged in 12 rows, she will have to remove 9 books from the arrangement. If she decides to have 13 books arranged in 13 rows, she will need 16 more books to fit into the arrangement. 
Example 2: What perfect sqaure number should Martin add to 153 so that the new number obtained is also a perfect square?
Solution:
153 lies between 144 and 169, which are the perfect squares closer to 153.
Adding 16 to 153,
153 + 16 = 169
√16 = 4
√169 = 13
16 is a perfect square number
169 is also a perfect square number. 
Example 3: What perfect sqaure number should Martin subtract from 153 so that the new number obtained is also a perfect square?
Solution:
153 lies between 144 and 169 which are the perfect squares closer to 153.
Subtracting 9 from 153,
153  9 = 144
√9 = 3
√144 = 12
9 is a perfect square number
144 is also a perfect square number.
FAQs on the Square Root of 153
What is the Value of the Square Root of 153?
The square root of 153 is 12.36931.
Why is the Square Root of 153 an Irrational Number?
Upon prime factorizing 153 i.e. 3^{2} × 17^{1}, 17 is in odd power. Therefore, the square root of 153 is irrational.
Evaluate 6 plus 13 square root 153
The given expression is 6 + 13 √153. We know that the square root of 153 is 12.369. Therefore, 6 + 13 √153 = 6 + 13 × 12.369 = 6 + 160.801 = 166.801
What is the Value of 10 square root 153?
The square root of 153 is 12.369. Therefore, 10 √153 = 10 × 12.369 = 123.693.
What is the Square Root of 153 in Simplest Radical Form?
We need to express 153 as the product of its prime factors i.e. 153 = 3 × 3 × 17. Therefore, √153 = √3 × 3 × 17 = 3 √17. Thus, the square root of 153 in the lowest radical form is 3 √17.
If the Square Root of 153 is 12.369. Find the Value of the Square Root of 1.53.
Let us represent √1.53 in p/q form i.e. √(153/100) = 1.53/10 = 1.237. Hence, the value of √1.53 = 1.237
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