Square Root of 150
The square root of 150 is expressed as √150 in the radical form and as (150)^{½} or (150)^{0.5} in the exponent form. The square root of 150 rounded up to 9 decimal places is 12.247448714. It is the positive solution of the equation x^{2} = 150. We can express the square root of 150 in its lowest radical form as 5 √6.
 Square Root of 150: 12.24744871391589
 Square Root of 150 in exponential form: (150)^{½} or (150)^{0.5}
 Square Root of 150 in radical form: √150 or 5 √6
What Is the Square Root of 150?
 The square root of 150 = 150^{½ }= √150
 √150 = √(a × a) which is √150 = √(12.247 × 12.247) or √(12.247 × 12.247) ⇒ √150 = ±12.247
 We know that on prime factorization, 150 = 2 × 3 × 5 × 5. Thus, in the simplest radical form √150= √(2 × 3 × 5 × 5) = 5√6
Is Square Root of 150 Rational or Irrational?
Irrational numbers are the real numbers that cannot be expressed as the ratio of two integers p/q. √150 = 12.24744871391589 and hence, the square root of 150 is an irrational number where the numbers after the decimal point go up to infinity.
How to Find the Square Root of 150?
The square root of 150 or any number can be calculated in many ways. Two of them are the approximation method and the long division method.
Square Root of 150 by Approximation Method
 Take two perfect square numbers which are just smaller than 150 and just greater than 150. √144 < √150 < √169
 12 < √150 < 13
 Divide 150 by 12 or 13.
 Let us divide by 13 ⇒ 150 ÷ 13 = 11.53
 Find the average of 11.53 and 13.
 (11.53 + 13) / 2 = 24.53 ÷ 2 = 12.265
 √50 ≈ 12.26
Square Root of 150 by the Long Division Method
The long division method helps us to find a more accurate value of the square root of any number. The following are the steps to evaluate the square root of 150 by the long division method.
 Step 1: Write 150.000000. Take the number in pairs from the right. We will have 50 as one pair and 1 stands alone. Now divide 1 by a number such that (number × number) gives ≤ 1.
 Obtain quotient = 1 and remainder = 0. Double the quotient. We get 2. We have 20 as our new divisor. Bring down 50 for division.
 Step 2: Find a number such that (20 + that number) × that number gives the product ≤ 50. We find 22 × 2 = 44. Subtract this from 50 and get the remainder as 6. Bring down two zeros. 600 is our new dividend.
 12 is our quotient. Double it. 240 is our new divisor. Find a number such that (240 + the number) × number gives 600 or less than that. We will consider 2 as the number. 242 × 2 = 484
 Step 3: Quotient is 12.2 and the remainder is 116. Bring down the next pair of zeros. 11600 becomes the new dividend.
 Double the quotient. 122 × 2 = 244. Have 2440 in the place of the new divisor. Find a number such that (2440 + that number) × number ≤ 11600.
 We find 2444 × 4 = 9776. Subtract this from 11600 and get the remainder 1824. Bring down 00.
 Repeat the steps until we approximate the square root to 3 decimal places. √150 = 12.247
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Think Tank
Did you know that on prime factorization of 150, we get: 2 × 3 × 5 × 5. Thus, 2 and 3 are the factors that don't have a pair. So 6 is the least number to be multiplied with 150 to make it a perfect square. 6 is the least number to be divided with 150 to make a perfect square. Can you find those perfect squares and their square roots?
Important Notes
 The square root of 150 is 12.247 approximated to 3 decimal places.
 The simplified form of √150 in its radical form is 5√6.
 √150 is an irrational number. It is a real number with 2 roots, i.e. √150 = ±12.247
Square Root of 150 Solved Examples

Example 1: How can we prove that 150 is not a perfect square?
Solution:
We know that the sum of n consecutive odd numbers = n2. Let us subtract 150 by consecutive odd numbers to check if the result is 0. If the repeated subtraction results in 0, it is a perfect square.
 150  1 = 149
 149  3 = 146
 146  5 = 141
 141  7 = 134
 134  9 = 125
 125  11 = 114
 114  13 = 101
 101  15 = 86
 86  17 = 69
 69  19 = 50
 50  21 = 29
 29  23 = 6
We can observe that the final result is not 0. Therefore, we can conclude that 150 is not a perfect square. We know that 6 has to be subtracted from 150 to make it a perfect square. 150  6 = 144; √144 = 12

Example 2: Evaluate √1.5
Solution:
√1.5 = √(150/100)
√150 = 12.247
√100 = 10
√(150/100) = 12.247 / 10 = 1.2247
√1.5 = 1.2247

Example: If the area of an equilateral triangle is 150√3 in^{2}. Find the length of one of the sides of the triangle.
Solution:
Let 'a' be the length of one of the sides of the equilateral triangle.
⇒ Area of the equilateral triangle = (√3/4)a^{2} = 150√3 in^{2}
⇒ a = ±√600 in
Since length can't be negative,
⇒ a = √600 = 2 √150
We know that the square root of 150 is 12.247.
⇒ a = 24.495 in
FAQs on the Square Root of 150
What is the Value of the Square Root of 150?
The square root of 150 is 12.24744.
Why is the Square Root of 150 an Irrational Number?
Upon prime factorizing 150 i.e. 2^{1} × 3^{1} × 5^{2}, 2 is in odd power. Therefore, the square root of 150 is irrational.
What is the Square Root of 150?
The square root of 150 is an imaginary number. It can be written as √150 = √1 × √150 = i √150 = 12.247i
where i = √1 and it is called the imaginary unit.
What is the Square of the Square Root of 150?
The square of the square root of 150 is the number 150 itself i.e. (√150)^{2} = (150)^{2/2} = 150.
What is the Square Root of 150 in Simplest Radical Form?
We need to express 150 as the product of its prime factors i.e. 150 = 2 × 3 × 5 × 5. Therefore, √150 = √2 × 3 × 5 × 5 = 5 √6. Thus, the square root of 150 in the lowest radical form is 5 √6.
Is the number 150 a Perfect Square?
The prime factorization of 150 = 2^{1} × 3^{1} × 5^{2}. Here, the prime factor 2 is not in the pair. Therefore, 150 is not a perfect square.