# Square root of 150

The square root of 150 is 12.247. Before we start how to compute the square root of 150, let us note down few interesting facts about the number 150.

150 is a composite even number.

150 is the sum of eight consecutive primes from 7 to 31, that is, 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 = 150.

Also Read:  Properties of square roots

#### Important Things:

• √150 = 12.247
• 150 is not a perfect square.
• 1501/2 is the exponential form of square root of 150.
• √150 is the radical form of square root of 150.
• The square root of 150 is 12.247 corrected up to 3 decimal places.
• Square of 150: 150×150 = 22500
• √150 is a quadratic surd.
• The simplified form of √150 is 5√6.

Let us now calculate the square root of 150.

#### Simplify square root of 150?

Let us now find the simplest radical form of the square root of 150. We will write 150 as the product of a perfect square and a non-perfect square. Note that

150=25×6.

Here 25 is a perfect square number whose square root is 5. We will take square root on both sides of 150=25×6. Doing that we get

$\sqrt{150}=\sqrt{25 \times 6}$

$=\sqrt{25} \times \sqrt{6}$  $[\because \sqrt{a \times b}=\sqrt{a} \times \sqrt{b}]$

$=5 \times \sqrt{6}=5\sqrt{6}$

So the simplest form of the square root of 150 is 5√6.

#### What is the value of root 150?

We have computed that the square root of 150 is 5√6. Using the fact √6=2.4494, we get that

√150 = 5√6 = 5 × 2.4494 = 12.247

So 12.247 is the value of the square root of 150.

#### Square root of 150 by Prime Factorization

To find the square root of 150 by the prime factorization method, we will first need to factorize the number 150.

As 150 is an even number, 2 will divide it. Thus we get

150=2×75.

Next, we will factorize 75. As the sum of digits of 75, that is, 7+5=12 is divisible by 3, the number 75 will be divisible by 3. So we have

75=3×25

At last, 25=5×5.

So the prime factorization of 150 is

150=2×3×5×5

Taking square root on both sides, we get that

$\sqrt{150}=\sqrt{2 \times 3 \times 5 \times 5}$

$=\sqrt{2 \times 3} \times \sqrt{5 \times 5}$  $[\because \sqrt{x \times y}=\sqrt{x} \times \sqrt{y}$ with $x=2 \times 3$ and $y=5 \times 5]$

$=\sqrt{6} \times 5$  $[\because \sqrt{a \times a}=a]$

$=5\sqrt{6}$

∴  the square root of 150 is 5√6.

#### Is Square root of 150 Rational?

From above we have √150=5√6. We all know that √6 is not a rational number. This implies that the square root of 150 is not a rational number. Note that √150 is a real number. Moreover, √150 is an irrational number.

#### Is 150 a perfect square number?

Since √150=5√6, we deduce that the square root of 150 is not a natural number. As a result, the number 150 is not a perfect square number. So its square root, that is, √150 is a quadratic surd.

Conclusion: 150 is a non-perfect square number.