Square root of 75: Simplified Radical Form

The simplified radical form of square root 75 is 5√3, that is, √75 = 5√3. The value of root 75 is equal to 8.66. Square root of 75 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 75^1/2
• 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 × √3

= 5√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 √75=5√3. As we know that √3=1.732, we obtain that

√75=5√3

= 5 × 1.732

= 8.66

So the value of the square root of 75 is 8.66

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Is Square Root of 75 Rational?

Recall that √75=5√3. As the square root of 3 is an irrational number, we conclude that the square root of 75 is an irrational number. So √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 × 15. In the same way, we have 15=5 × 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 × √3

= 5√3

So 5√3 is the value of the square root 75, and this is obtained by the prime factorisation method.

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