The square root of 45 is 6.708. We denote it by the symbol √45. In this section, we will learn about √45.

Table of Contents

**Key Things**

- 45 is a composite odd number.
- √45=6.708
- Square of 45: 45×45 = 2025
- The radical form of square root of 45 is √45.
- √45 is a quadratic surd.
- The simplest radical form of square root of 45 is 3√5.
- The exponential form of square root of 45 is 45
^{1/2}. - Square root of 45 is 6.708 corrected up to 3 decimal places
- 45 is not a perfect square number.
- Square root of 45 is an irrational number.

Let us now calculate the square root of 45.

**Simplify square root of 45**

We will now find the simplest radical form the square root of 45. Note the 45 is a non-perfect square number. Thus we will try to express 45 as a product of a perfect square number and a non-perfect square number. See that

45 = 9×5.

Here 9 is a perfect square number whose square root is 3.

Taking square root on both sides of 45 = 9×5, we get that

√45 = √(9×5)

= √9 × √5 as we know that √(a×b)=√a × √b

= 3 × √5 as √9=3

= 3√5

∴ the simplest form of the square root of 45 is 3√5.

#### What is the value of root 45?

From above we see that the square root of 45 is 3√5.

We know that √5=2.236 approximately. Thus we obtain that

√45 = 3√5 = 3 × 2.236 = 6.708

So the value of the square root of 45 is 6.708

#### Square root of 45 by Prime Factorization

To find the square root of 45 by the prime factorization method, we need to factorize the number 45.

As the sum of digits of 45, that is 4+5=9 is divisible by 3, we obtain that 45 is divisible by 3. So we have

45 = 3×15.

Now since 15=3×5, we obtain that 45=3×15=3×3×5. Thus,

45 = 3×3×5 …(∗)

is the prime factorization of 45 as both 3 and 5 are prime numbers.

Taking square root on both sides of (∗), we get

√45 = √(3×3×5)

= √(3×3) × √5 as we know that √(a×b)=√a×√b. Here a=3×3 and b=5.

= 3 × √5 as √(a×a)=a

= 3√5

So 3√5 is the value of the square root of 45.

#### Is Square root of 45 Rational?

From above we have that √45=3√5. It is known that √5 is an irrational number.

⇒ the square root of 45 is an irrational number.

In other words, √45 is not a rational number.

#### Is 45 a perfect square?

As √45=3√5, we deduce the square root of 45 is not a natural number.

⇒ the number 45 is not a perfect square number.

⇒ 45 is a non-perfect square number.

Note that √45 is a quadratic surd.

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