Square Root of 135

The square root of 135 is ±11.618. Finding the square root of a number is the inverse process of finding the perfect square. The square root of 135 is written as √135. 
 

The different ways to find the square root of a number are prime factorization, long division  and approximation/estimation method
 

The prime factorization of 135 breaks 135 into its prime numbers. 

The numbers 3 and 5 are the prime numbers 

Prime factorization of 135 is 3³ × 5

Since 3 and 5 are not repeating, we can’t pair them

Therefore, √135 is expressed as 3√3√5, the simplest radical form
 

The long division method finds the square root of non-perfect squares.

Step 1: Write down the number 135

Step 2: Number 135 is a three-digit number, so pair them as (1), (35)

Step 3: Find the largest that is closest to the first pair (1), which is 1²

Step 4: Write down 1 as the quotient, which will be the first digit of the square root.

Step 5: Subtracting 1²  from 1 will leave zero as the remainder. Now bring down the second pair (35) and place it beside 0.

Step 6: Now double the quotient you have, that is multiply the quotient 1 with 2 and the result will be 2

Step 7: Choose a number such that it can be placed after 2. The two-digit number created should be less than the second pair (35). Here, we place number 1 after 2, because the number formed is less than 35.

Step 8: Subtract 21 from 35  → 35-21 = 14. Now add a decimal point after the new quotient and adding two zeros will make it 1400

Step 9: Apply step 7 over here and continue the process until you reach 0.

Step 10: We can write  √135 as 11.618

The approximation method finds the estimated square root of non-perfect squares.

Step 1: Identify the closest perfect square to 135. Numbers 121 and 144 are the closest perfect square to 135.

Step 2: We know that  √121 = 11 and  √144 = 12. Thus, we can say that  √135 lies between  11 and 12.

Step 3: Check if  √135 is closer to 11 or 12. Let us take 11.5 and 12. Since (11.5)² is 132.25 and (12)² is 144,  √135 lies between them.

Step 4: We can keep changing the values of 11.5 to 11. 6 and iterate the same process without changing 12 as the closest perfect square root.

The result of  √135 will be 11.616
 

• Perfect Square: Product obtained when the same number gets multiplied twice

• Approximate Value: Value closer to the exact number

• Prime Factorization: Breaking down the number into its prime factors.