Square Root of 135
Did you know 135 is expressed as the sum of the consecutive powers of its digits, that is 135 = 1^{3} + 3^{3} + 5^{3}? The square root of 135 can be written as √135. This is raising 135 to the power ½. In this mini lesson, let us will learn about the square root of 135 and learn how to find the square root of 135 by long division method.
 Square Root of 135: √135 = 11.618
 Square of 135: 135^{2} = 18,225
What Is the Square Root of 135?
 The square root of a number is the inverse process of squaring a number. A square of a number x is y^{2 }⇒ x = y × y.
 √123 = √(11.618 × 11.618) or √(11.618 × 11.618).
 Thus we have 135^{½} ⇒ √135 = ±11.618.
Is Square Root of 135 Rational or Irrational?
 √135 = 11.61895003862225.
 The value of the square root of 135 is a decimal number whose digits after the point are neverending or nonterminating.
 √135 cannot be written in the form of p/q, hence, it is an irrational number.
Tips and Tricks
 The square root of 135 lies between the square root of 121 and 144. Therefore, 11 < √135 < 12.
 You can then use the average method to evaluate the approximate value of √135. Divide 135 by 12. We get 11.25. Find the average of 11.25 and 12. We get the approximate value of √135 as 11.625.
How to Find the Square Root of 135?
The square root of 135 or any number can be calculated in many ways. Two of the common methods are the prime factorization method and the long division method.
Square Root of 135 by Prime Factorization Method
 Prime factorization is expressing 135 as a product of its prime factors as 135 = 3 × 3 × 3 × 5.
 Hence, √135 = √(3 × 3 × 3 × 5) = √(3^{2 }× 15)^{ }= √3^{2 }× √15.
 In the exponential form, 135^{½} = (3^{2 })^{½ }× 15^{½} = (3)^{ }× 15^{½}.
 √135 ^{ }= 3^{ }× √15 ⇒ √135 ^{ }= 3^{ }√15.
Square Root of 135 by the Long Division Method
The long division method helps us to find a more accurate value of the square root of any number. Let's see how to find the square root of 123 by the long division method.
 Step1: Write 135 as 1 35. 00 00 00 and from the right take the numbers in pairs. 1 is left alone. Let's divide 1 first.
 Step 2: Find a number × number such that it results ≤ 1. We get 1 × 1 = 1.
 Step 3: Get the remainder as 0 and bring down the next pair 35 for division.
 Step 4: Double the quotient obtained. It is 2 and we hav 20 as our new divisor.
 Step 5: Find a (number + 20 ) × number such that it results ≤ 35. We get (number + 20 ) × number = (1 + 20) × 1 = 21.
 Step 6: Subtract this from 35. Get the remainder as 14. Bring down the next pair of zeros. 14 00 is taken for division.
 Step 7: Double the quotient obtained. It is 22 and we hav 220 as our new divisor.
 Step 8: Find a (number + 220 ) × number such that it results ≤ 14 00. We get (6+ 220 ) × 6 = 13 56
 Step 9: Subtract this from 14 00 and get the remainder as 44. Bring down the next pair of zeros. 44 00 is our new divisor.
 Step 10: Quotient we have got so far is 11.6 and on doubling we get 232. We take 2320 as our new divisor.
 Step 11: Find a (number + 2320 ) × number such that it results ≤ 44 00. We get (number 1 + 2320 ) × 1 = (1 + 2320 ) × 1 = 23 21
 Step 12: We repeat the process until we approximate the square root of 135 to 3 decimal places.
Explore square roots using illustrations and interactive examples:
Important Notes
 The square root of 135 is 11.618 approximated to 3 decimal places.
 The simplified form of √135 in its radical form is 3√15.
 √135 is an irrational number.
Square Root of 135 Solved Examples

Example 1: What could be the smallest number multiplied to 135 to make it a perfect square? Find the square root of the number so obtained.
Solution:
On prime factorization we express 135 as 135 = 3 × 3 × 3 × 5. Here 3 and 5 don't have a pair. Thus, 15 has to be multiplied with 135 to make it a perfect square.
3 × 3 × 3 × 5 × 3 × 5 ⇒ 3 × 3 × 3 × 3 × 5 × 5 = 2025.
Hence, √2025 = 45. 
Example 2: If 135 plants are to be planted in equal number of rows and columns, then how many plants are out of this arrangement?
Solution:
Rows × columns = 135
Given that rows = columns
rows × columns = n × n = n^{2}
n^{2 }= 135
We cannot find a number n that is not a whole number. The nearest perfect square lesser than 135 is 121.
Thus, 11 × 11 = 121.
135  121 = 14.
Thus, 14 plants are out of this arrangement.
FAQs on Square Root of 135
What is the square root of 135?
The square root of 135 is 11.618.
What is the square root of 135 simplified?
3√15 is the simplest form of √135.
How to find the square root of 135?
The square root of 135 can be found using the long division method.
Is √135 a rational number?
√135 is an irrational number because the value of √135 is a nonterminating decimal.
How to find the square root of 135 to the nearest hundredth?
The square root of 135 is evaluated using the long division method and rounded off to the nearest hundredth as √135 = 11.6.
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