# Square root of 75

In this section, we will discuss about the square root of 75. It is a number when multiplied by itself will produce the number $75.$  The square root of 75 is denoted by √75. Few things to remember:

• Note that 75 is an odd composite number.
• 75 is not a perfect square.
• The value of square root of 75 is 8.660254…
• 75 square: 752 = 75×75 = 5625
• The radical form of square root of 75 is √75.
• The exponential form of square root of 75 is 751/2
• Square root of 75 is 8.6603 corrected up to four decimal places.
• Square root of 75 is not a rational number.
• The simplest radical form of √75 is 5√3.
• Note that √75 is a quadratic surd.

#### Simplify Square Root of 75

What is the simplest radical form of  square root of 75? Note that 75 = 25×3. We will take square root on both sides.

∴ $\sqrt{75}=\sqrt{25 \times 3}$

$=\sqrt{25} \times \sqrt{3}$ $[\because \sqrt{x \times y}=\sqrt{x} \times \sqrt{y}]$

$=5 \times \sqrt{3}$

$=5\sqrt{3}$

So the simplified form of the square root of 75 is 5√3.

#### What is the Square Root of 75

From above we have that $\sqrt{75}=5\sqrt{3}.$ As we know that $\sqrt{3}=1.732,$ we obtain that

$\sqrt{75}=5\sqrt{3}$

$=5 \times 1.732$

$=8.66$

So the value of the square root of $75$ is $8.66$

#### Is Square Root of 75 Rational?

Recall that $\sqrt{75}=5\sqrt{3}.$ As the square root of $3$ is an irrational number, we conclude that the square root of $75$ is an irrational number. So $\sqrt{75}$ is not a rational number.

#### Square Root of 75 by Prime Factorization

Using the prime factorization method to compute the square root of $75,$ we need to first factorize $75.$ As $75$ has unit digit $5,$ it will be divisible by $5,$ so we can write $75=5 \times 15.$ In the same way, we have $15=5 \times 3.$ So finally we get the prime factorization of $75$ which is

$75=5 \times 5 \times 3$

Taking square root on both sides, we get that

$\sqrt{75}=\sqrt{5 \times 5 \times 3}$

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

$=5 \times \sqrt{3}$

$=5\sqrt{3}$

So $5\sqrt{3}$ is the value of the square root $75,$ and this is obtained by the prime factorization method.

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