Square Root of 150

The square root of 150 is irrational. The approximate value of square root of 150 is 12.25. If there exists a number “n” such that “n × n” = 150, then “n” will be called the square root of 150 and 150 will be termed the perfect square number. Note that “n” can be an integer. The methods to find “n” are either by prime factorisation method, long division method or repeated subtraction method. All the details on these methods are detailed in this article. 

Consider the number 12 × 12 = 144 and 13 × 13 = 169. 150 lies between 144 and 169. So the square root of 150 lies between 12 and 13, as 150 is not a perfect square number. Hence, the square root of 150 is also not a perfect integer. Therefore, √150 is an irrational number. You can refer to the 1 to 50 Square and Square root List.

Note the Following:

• The Square root of 150 = 2√150 = √150 where ‘√’ is the radical, and 150 is the radicand and 2 is the index.
• Exponential Form of Square root of 150 = 150^1/2
• Solution for √150 ≈12.24
• The square root of 150 is Irrational = True

What is the Square root of 150?

The square root of 150 is12.24, as the square root of 150 is irrational.

How to Find the Square root of 150?

There are three methods to find the Square root of 150 

• Prime Factorisation method
• Long Division method
• Repeated Subtraction method

Square root of 150 by Prime Factorisation Method

In the prime factorization method, 150 is divided by prime factors. Starting with 2. 

The prime factors of 150 are 2, 3 and 5. 

150 = 2 × 3 × 5 × 5

If we had to group the same prime numbers of two, then only 5 can be grouped.

150 = 2 × 3 × (5 × 5)

√150 = √(2 × 3 × (5 × 5))

√150 = 5√(2 × 3) = 5√6

The division of 150 using prime factors is given below;

Square root of 150 by Long Division Method

To understand the long division method, follow the below mentioned detailed steps;

Step 1: Group the given number into pairs. 

Step 2: Divide the first pair, using the divisor such that the product is less than or equal to.

Step 3: Continue the division by bringing the next pair.

Step 4: The first divisor in the next division to be  (Divisor of first division + quotient of first division) such that:

d1 × 1 =

d2 × 2 =

d3 × 3 =

 And so on. 

Step 5: If the division is complete with the remainder as zero, then the quotient will be the square root. Else the number is not a perfect square number.

Applying the above steps to 150.

Pairing 1 and 50

The squares are:

1 × 1 = 1

No further division is possible and the remainder is non-zero. Hence 150 does not have a perfect square. Therefore 150 is an irrational number which is 5√6.

Square root of 150 by Repeated Subtraction Method

When 150 is subtracted by odd numbers, i.e., 150 – 1 = 149. Now 149 – 3 = 146 then with 7 and so on. When this repeated subtraction results in zero at any point, 150 will be a perfect square number. If the subtraction results in a negative number, then 150 will not be a perfect square number, and hence it won’t have a square root. 

The table below shows repeated subtraction for the given number 150. 

Since the difference does not become zero, 150 is not a perfect square number and hence it does not have a perfect square root. The root lies between 12 and 13, resulting in a decimal root. 

Video Lessons

Visualising square roots

Finding Square roots

Related Articles

• Square Root
• What is Square root?
• Finding Square roots
• Finding Square root with Example
• Square root Calculator
• Squares and Square roots.

Solved Examples

1. Which number is that lesser than or greater than 150 that has a perfect square root?

144, which is less than 150 and 169, which is greater than 150, are the perfect square numbers. 12 and 13 are their perfect square roots, respectively.

2. What is the square root of 169?

13 is the square root of 169.