## Square Root of a Perfect Square by using the Prime Factorization Method

To find the square root of a perfect square by using the prime factorization method when a given number is a perfect square:

Step I: Resolve the given number into prime factors.

Step II: Make pairs of similar factors.

Step III: Take the product of prime factors, choosing one factor out of every pair.

** Examples on square root of a perfect square by using the prime factorization method;**

*1. Find the square root of 484 by prime factorization method.*

Solution:

Resolving 484 as the product of primes, we get

484 = 2 × 2 × 11 × 11

√484 = √(__2 × 2__ × __11 × 11__)

= 2 × 11

**Therefore, √484 = 22**

*2. Find the square root of 324.*

Solution:

The square root of 324 by prime factorization, we get

324 = 2 × 2 × 3 × 3 × 3 × 3

√324 = √(__2 × 2__ × __3 × 3__ × __3 × 3__)

= 2 × 3 × 3

**Therefore, √324 = 18**

*3. Find out the square root of 1764.*

Solution:

The square root of 1764 by prime factorization, we get

1764 = 2 x 2 x 3 x 3 x 7 x 7.

√1764 = √(__2 x 2__ x __3 x 3__ x __7 x 7__)

= 2 x 3 x 7

**Therefore, √1764 = 42.**

*4. Evaluate √4356*

Solution:

By using prime factorization, we get

4356 = 2 x 2 x 3 x 3 x 11 x 11

√4356 = √(__2 x 2__ x __3 x 3__ x __11 x 11__)

= 2 × 3 × 11

**Therefore, √4356 = 66.**

*5. Evaluate √11025*

Solution:

By using prime factorization, we get

11025 = 5 x 5 x 3 x 3 x 7 x 7.

√11025 = √(__5 x 5__ x __3 x 3__ x __7 x 7__)

= 5 × 3 × 7

**Therefore, √11025 = 105**

*6. In an auditorium, the number of rows is equal to the number of chairs in each row. If the capacity of the auditorium is 2025, find the number of chairs in each row.*

Solution:

Let the number of chairs in each row be x.

Then, the number of rows = x.

Total number of chairs in the auditorium = (x × x) = x^{2}

But, the capacity of the auditorium = 2025 (given).

Therefore, x^{2} = 2025

= __5 × 5__ × __3 × 3__ × __3 × 3__

x = (5 × 3 × 3) = 45.

**Hence, the number of chairs in each row = 45**

*7. Find the smallest number by which 396 must be multiplied so that the product becomes a perfect square.*

Solution:

By prime factorization, we get

396 = __2 × 2__ × __3 × 3__ × 11

It is clear that in order to get a perfect square, one more 11 is required.

So, the given number should be multiplied by 11 to make the product a perfect square.

*8. Find the smallest number by which 1100 must be divided so that the quotient is a perfect square.*

Solution:

Expressing 1100 as the product of primes, we get

1100 = 2 × 2 × 5 × 5 × 11

Here, 2 and 5 occur in pairs and 11 does not.

Therefore, 1100 must be divided by 11 so that the quotient is 100

i.e., 1100 ÷ 11 = 100 and 100 is a perfect square.

*9. Find the least square number divisible by each one of 8, 9and 10.*

Solution:

The least number divisible by each one of 8, 9, 10 is their LCM.

Now, LCM of 8, 9, 10 = (2 × 4 × 9 × 5) = 360

By prime factorization, we get

360 = 2 × 2 × 2 × 3 × 3 × 5

To make it a perfect square it must be multiplied by (2 × 5) i.e., 10.

**Hence, the required number = (360 × 10) = 3600.**

**Square Root**

**Square Root**

**Square Root of a Perfect Square by using the Prime **

Factorization Method

**Square Root of a Perfect Square by Using the Long**

Division Method

**Square Root of Numbers in the Decimal Form**

**Square Root of Number in the Fraction Form**

**Square Root of Numbers that are Not Perfect Squares**

**Table of Square Roots**

**Practice Test on Square and Square Roots**

**Square Root- Worksheets****Worksheet on Square Root using Prime Factorization**

Method

**Worksheet on Square Root using Long Division Method**

**Worksheet on Square Root of Numbers in Decimal and**

Fraction Form

8th Grade Math Practice

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