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 
 

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

• Prime factorization method

• Long division method

• Approximation/Estimation 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

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.

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

• 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