Square Root of 189

The square root of 189 is irrational. The approximate value of square root of 189 is 13.75. 

Consider the number 25. The number 25 can be expressed as a product of 5 and 5, i.e., 5 × 5 = 25. Here 25 is the perfect square number and 5 is its square root. Similarly, we need to find a number such that when the same number is multiplied by itself, the product should be 189. That number becomes the square root of 189. The mathematical notation would be “n × n = 189”, where “n” is the square root. Note that “n” can be an integer. The methods to find “n” are either by prime factorisation method, long division method or repeated subtraction method. You can refer to the Finding Square rootsfor more details.

Note the Following:

• The Square root of 189 = 2√189 = √189 where ‘√’ is the radical, and 189 is the radicand and 2 is the index.
• Exponential Form of Square root of 189 = 189^1/2
• Solution for √189 = 3√21≈ 13.75
• The square root of 189 is Irrational = True

What is the Square root of 189?

The square root of 189 is 13.75, as the square root of 189 is irrational.

How to Find the Square root of 189?

There are three methods to find the Square root of 189 

• Prime Factorisation method
• Long Division method
• Repeated Subtraction method

Square root of 189 by Prime Factorisation Method

The steps followed In the prime factorisation method are

1. Divide the given number by prime factors.
2. Each group should have the same two factors. 
3. 1 number from the group to be removed from the root. to get the square root. 
4. Product of removed prime factors from the root forms the square root. 

Applying the above steps for 189 are:

189 = 3 × 3 × 3 × 7

√189 = √(3 × 3 × 3 × 7) = 3 √3 × 7 = 3 √21

189 is irrational and it does not have a perfect square root.

Square root of 189 by Long Division Method

To understand the long division method, follow the below mentioned detailed steps;

Step 1: Group the given number into pairs. 

Step 2: Divide the first pair, using the divisor such that the product is less than or equal to.

Step 3: Continue the division by bringing the next pair.

Step 4: The first divisor in the next division to be (Divisor of first division + quotient of first division) such that:

d1 × 1 =

d2 × 2 =

d3 × 3 =

 And so on. 

Step 5: If the division is complete with the remainder as zero, then the quotient will be the square root. Else the number is not a perfect square number.

Applying the above steps to 189.

Pairing 1 and 89.

The squares are: 

1 × 1 = 1

2 × 2 = 4

Hence 189 does not have a perfect square. Therefore 189 is an irrational number whose root lies between 13 and 14.

Square root of 189 by Repeated Subtraction Method.

In the repeated Subtraction method, 189 is subtracted by odd numbers to the difference obtained from the previous subtraction steps. If the difference at any point is zero, then 189 becomes the perfect square number and the step at which it becomes zero is its square root. If the difference is negative, then the number 189 is not a perfect square number and its root will be irrational. 

Let us find out if 189 is a perfect square or not with the difference table 

At step number 14, the difference becomes negative and not zero. Hence 189 is irrational and the root lies between 13 and 14. 

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Solved Examples

1. Does 189 have a perfect square root?

Solution: No. 189 does not have a perfect square root.

2. What is the square of 189?

Solution: 189 square is 35721.