Square Root of 169
2026-02-28 23:58 Diff
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Last updated on August 5, 2025

The square root of 169 is a value “y” such that when “y” is multiplied by itself → y ⤫ y, the result is 169. The number 169 has a unique non-negative square root, called the principal square root.

What Is the Square Root of 169?

The square root of 169 is ±13, where 13 is the positive solution of the equation x2 = 169. Finding the square root is just the inverse of squaring a number and hence, squaring 13 will result in 169. The square root of 169 is written as √169 in radical form, where the ‘√’  sign is called the “radical” sign. In exponential form, it is written as (169)1/2 
 

Finding the Square Root of 169

We can find the square root of 169 through various methods.They are:

  • Prime factorization method
  • Approximation/Estimation method
     

Square Root of 169 By Prime Factorization Method

The prime factorization of 169 can be found by dividing the number by prime numbers and continuing to divide the quotients until they can’t be separated anymore, i.e., we first prime factorize 169 and then make pairs of two to get the square root.

So, Prime factorization of 169 = 13 × 13


Square root of 169= √[13 × 13] = 13

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Square Root of 169 By Long Division Method

This method is used for obtaining the square root for non-perfect squares, mainly. It usually involves the division of the dividend by the divisor, getting a quotient and a remainder too sometimes.


Follow the steps to calculate the square root of 169:


 Step 1: Write the number 169 and draw a bar above the pair of digits from right to left.


Step 2: Now, find the greatest number whose square is less than or equal to 1. Here, it is 1 because 12=1 < =1


Step 3: now divide 169 by 1 (the number we got from Step 2) such that we get 1 as a quotient, and we get a remainder.  Double the divisor 1, we get 2, and then the largest possible number A1=3 is chosen such that when 3 is written beside the new divisor 2, a 2-digit number is formed →23, and multiplying 3 with 23 gives 69, which when subtracted from 69, gives 0. Repeat this process until you reach the remainder of 0. 

  Step 4: The quotient obtained is the square root of 169. In this case, it is 13.

Square Root of 169 By Subtraction Method

We know that the sum of the first n odd numbers is n2. We will use this fact to find square roots through the repeated subtraction method. Furthermore, we just have to subtract consecutive odd numbers from the given number, starting from 1. The square root of the given number will be the count of the number of steps required to obtain 0. Here are the steps:


Step 1: take the number 169 and then subtract the first odd number from it. Here, in this case, it is 169-1=168


Step 2: we have to subtract the next odd number from the obtained number until it comes zero as a result. Now take the obtained number (from Step 1), i.e., 168, and again subtract the next odd number after 1, which is 3, → 168-3=165. Like this, we have to proceed further.


Step 3: now we have to count the number of subtraction steps it takes to yield 0 finally. Here, in this case, it takes 13 steps

 
So, the square root is equal to the count, i.e., the square root of 169 is ±13

Common Mistakes and How to Avoid Them in the Square Root of 169

When we find the square root of 169, we often make some key mistakes, especially when we solve problems related to that. So, let’s see some common mistakes and their solutions.

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Problem 1

Find √(169⤬144) ?

Okay, lets begin

√(169⤬144)

= 13 ⤬12

= 156

Answer : 156
 

Explanation

 firstly, we found the values of the square roots of 169 and 144, then multiplied the values.

Well explained 👍

Problem 2

What is √169 multiplied by 13 ?

Okay, lets begin

√169 ⤬ 13

= 13⤬13

= 169


Answer: 169 
 

Explanation

finding the value of √169 and multiplying by 13.
 

Well explained 👍

Problem 3

Find the radius of a circle whose area is 169π cm².

Okay, lets begin

 Given, the area of the circle = 169π cm2


Now, area = πr2 (r is the radius of the circle)


So, πr2 = 169π cm2


We get, r2 = 169 cm2


r = √169 cm


Putting the value of √169 in the above equation, 


We get, r = ±13 cm


Here we will consider the positive value of 13.


Therefore, the radius of the circle is 13 cm.


Answer: 13 cm.
 

Explanation

 We know that, area of a circle = πr2 (r is the radius of the circle). According to this equation, we are getting the value of “r” as 13 cm by finding the value of the square root of 169
 

Well explained 👍

Problem 4

Find the length of a side of a square whose area is 169 cm²

Okay, lets begin

 Given, the area = 169 cm2


We know that, (side of a square)2 = area of square


Or,  (side of a square)2 = 169


Or,  (side of a square)= √169


Or, the side of a square = ± 13.


But, the length of a square is a positive quantity only, so, the length of the side is 13 cm.


Answer: 13 cm
  

Explanation

We know that, area of a circle = πr2 (r is the radius of the circle).According to this equation, we are getting the value of “r” as 13 cm by finding the value of the square root of 169
 

Well explained 👍

Problem 5

Find the length of a side of a square whose area is 169 cm²

Okay, lets begin

Given, the area = 169 cm2


We know that, (side of a square)2 = area of square


Or,  (side of a square)2 = 169


Or,  (side of a square)= √169


Or, the side of a square = ± 13.


But, the length of a square is a positive quantity only, so, the length of the side is 13 cm.


Answer: 13 cm
 

Explanation

We know that, (side of a square)2 = area of square. Here, we are given with the area of the square, so, we can easily find out its square root because its square root is the measure of the side of the square.
 

Well explained 👍

Problem 6

Find √169 / √100

Okay, lets begin

√169/√100

= 13/10

 = 1.3

Answer : 1.3 
 

Explanation

 we firstly found out the values of √169 and √100, then divided  .

Well explained 👍

FAQs on Square Root of 169

1.What is 169 divisible by?

169 is divisible by 1,13, and 169. 
 

2.How to factorize 168?

 168 = 2⤬2⤬2⤬3⤬7 ← this is the prime factorization of 168. We can factorize 168 by the prime numbers 2,3, and 7.
 

3.Is 169 a perfect square or non-perfect square?

169 is a perfect square, since 169 =(13)2.
 

4.Is the square root of 169 a rational or irrational number?

The square root of 169 is ±13. So, 13 is a rational number, since it can be obtained by dividing two integers and can be written in the form p/q, where p and q are integers.
 

5.What is the LCM of 26 and 169?

338 is the LCM of 26 and 169, where 338 is the smallest number that is a common multiple of both 26 and 169, and it is also the number that both 26 and 169 divide into evenly.
 

Important Glossaries for Square Root of 169

  • Exponential form:  An algebraic expression that includes an exponent. It is a way of expressing the numbers raised to some power of their factors. It includes continuous multiplication involving base and exponent. Ex: 2 × 2 × 2 × 2 = 16 Or, 24 = 16, where 2 is the base, 4 is the exponent 
  • Prime Factorization:  Expressing the given expression as a product of its factors. Ex: 48=2 × 2 × 2 × 2 × 3
  • Prime Numbers: Numbers which are greater than 1, having only 2 factors as →1 and Itself. Ex: 1,3,5,7,....
  • Rational numbers and Irrational numbers: The Number which can be expressed as p/q, where p and q are integers and q not equal to 0 are called Rational numbers. Numbers which cannot be expressed as p/q, where p and q are integers and q not equal to 0 are called Irrational numbers. 
  • Perfect and non-perfect square numbers: Perfect square numbers are those numbers whose square roots do not include decimal places. Ex: 4,9,25 Non-perfect square numbers are those numbers whose square roots comprise decimal places. Ex :3, 8, 24

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Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

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: He loves to play the quiz with kids through algebra to make kids love it.