Square Root of 45

In Mathematics, the square root of 45 is a number, which when multiplied by itself results in the original number 45. The square root of 45 is not a rational number as it cannot be expressed in the form of p/q. The two different methods to find the value of the square root of 45 are prime factorization and the long division method. Here, we will discuss in detail about the two different methods and the square root of 45 in both decimal and radical form with complete explanation.

Table of Contents:

• What Is The Square Root Of 45?
• Square Root Of 45 In Radical Form
• Square Root Of 45 By Prime Factorization Method
• Square Root Of 45 By Long Division Method
• Solved Examples
• FAQs

What is the Square Root of 45?

The square root of 45 is defined as a number, if it is multiplied by itself and gives the result as 45. The square root of 45 is symbolically expressed as √45.

Hence, √45 = √(Number × Number)

Thus, if we multiply the number 6.708 two times, we get the original value 45.

(i.e) √45 = √(6.708× 6.708)

√45 = √(6.708)²

Now, remove square and square root, we get

√45 = ± 6.708 (Rounded to three decimal places)

Square Root of 45 in Radical Form

The square root of 45 can also be represented in the radical form. To find the radical form of the square root of 45, we must know the prime factorization of 45. Thus, the prime factorization of 45 is 3² × 5 or 3 × 3 × 5. Thus, the square root of 45 in radical form is expressed as follows:

√45 = √(3 × 3 × 5)

√45 = √(5) ×√(3)²

√ 45 = 3√5.

Therefore, the square root of 45 in radical form is 3√5.

Square Root of 45 by Prime Factorization Method

The square root of 45 is found using two different methods. They are:

• Prime Factorization Method
• Long Division Method

Now, let us discuss the prime factorization method to find the square root of 45. To find the square root of 45 using the prime factorization method, one must know the prime factorization of 45. Thus, the prime factorization of 45 is 3× 3×5

Thus, √45 = √(3 × 3 × 5) = 3√5

We know that the value of √5 is 2.236.

Now, substitute the value in the above expression, we get

√45 = 6.708

√45 = 6.708 (approximately)

Hence, the square root of 45 in decimal form is approximately equal to 6.708 (rounded to three decimal places).

Square Root of 45 by Long Division Method

The steps to find the square root of 45 using the long division method is given as follows:

Step 1: Write the number 45 in decimal form. To find the exact value of the square root of 45, add 6 zeros after the decimal point. Hence, 45 in decimal form is 45.000000. Now, pair the number 45 from right to left by putting the bar on the top of the number.

Step 2: Now, divide the number 45 by a number, such that the product of the same number should be less than or equal to 45. Thus, 6×6 =36, which is less than 45. Thus, we obtained the quotient = 6 and remainder = 9.

Step 3: Double the quotient value, so we get 12, and assume that 120 is the new divisor. Now, bring down the value 00 for division operation. So, the new dividend obtained is 900. Now, find the number, such that (120 + new number) × new number should give the product value, which should be less than or equal to 900. Hence, (120+7) × 7 = 889, which is less than 900.

Step 4: Now subtract 889 from 900, and we get 11 as the new reminder, and 67 as a quotient.

Step 5: The new quotient obtained is 67, and double that. Hence, we get 134 and assume that 1340 is our new divisor. Now, bring down the two zeros and, we have 1100 as the new dividend.

Step 6: Find the number, such that (1340 + new number) × new number should give the product value that should be less than or equal to 1100. Thus, (1340+0)× 0 = 0, which is less than 1100.

Step 7: Subtract 0 from 1100, and we get 1100 as the new reminder.

Step 8: Continue this process until we get the approximate value of the square root of 45 upto three decimal places. (Note: keep the decimal point in the quotient value after bringing down all the values in the dividend).

Step 9: Thus, the approximate value of the square root of 45, √45 is 6.708.

Video Lessons on Square Roots

Visualising square roots

Finding Square roots

Examples

Example 1:

Simplify (√5√45) + 7

Solution:

Given: (√5√45) + 7…(1)

We know that the square root of 45 in radical form is 3√5.

Now, substitute √45 = 3√5 in (1) we get,

(√5√45) + 7= (√5. 3√5)+7

(√5√45) + 7= 15 + 7

(√5√45) + 7= 22

Therefore, (√5√45) + 7= 22

Example 2:

Find the value of a, if 3a + √45 = 8

Solution:

Given equation: 3a + √45 = 8…(1)

We know that √45 = 6.708.

Now, substitute the value in equation (1),

3a + 6.708 = 8

3a = 8 – 6.708

3a = 1.292

a = 1.292/3

a = 0.431

Therefore, the value of a is 0.431.