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Square Root of 189

The square root of 189 is expressed as √189 in the radical form and as (189)½ or (189)0.5 in the exponent form. The square root of 189 rounded up to 7 decimal places is 13.7477271. It is the positive solution of the equation x2 = 189. We can express the square root of 189 in its lowest radical form as 3 √21.

  • Square Root of 189: 13.74772708486752
  • Square Root of 189 in exponential form: (189)½ or (189)0.5
  • Square Root of 189 in radical form: √189 or 3 √21

Square Root of 189

1. What is the Square Root of 189?
2. How to find the Square Root of 189?
3. Is the Square Root of 189 Irrational?
4. FAQs

What is the Square Root of 189?

The square root of 189, (or root 189), is the number which when multiplied by itself gives the product as 189. Therefore, the square root of 189 = √189 = 3 √21 = 13.74772708486752.

☛ Check: Square Root Calculator

How to Find Square Root of 189?

Square Root of 189 by Long Division Method

Value of √189 by Long Division Method

Explanation:

  • Forming pairs: 01 and 89
  • Find a number Y (1) such that whose square is <= 1. Now divide 01 by 1 with quotient as 1.
  • Bring down the next pair 89, to the right of the remainder 0. The new dividend is now 89.
  • Add the last digit of the quotient (1) to the divisor (1) i.e. 1 + 1 = 2. To the right of 2, find a digit Z (which is 3) such that 2Z × Z <= 89. After finding Z, together 2 and Z (3) form a new divisor 23 for the new dividend 89.
  • Divide 89 by 23 with the quotient as 3, giving the remainder = 89 - 23 × 3 = 89 - 69 = 20.
  • Now, let's find the decimal places after the quotient 13.
  • Bring down 00 to the right of this remainder 20. The new dividend is now 2000.
  • Add the last digit of quotient to divisor i.e. 3 + 23 = 26. To the right of 26, find a digit Z (which is 7) such that 26Z × Z <= 2000. Together they form a new divisor (267) for the new dividend (2000).
  • Divide 2000 by 267 with the quotient as 7, giving the remainder = 2000 - 267 × 7 = 2000 - 1869 = 131.
  • Bring down 00 again. Repeat above steps for finding more decimal places for the square root of 189.

Therefore, the square root of 189 by long division method is 13.7 approximately.

Is Square Root of 189 Irrational?

The actual value of √189 is undetermined. The value of √189 up to 25 decimal places is 13.74772708486752001976414. Hence, the square root of 189 is an irrational number.

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Square Root of 189 Solved Examples

  1. Example 1: Solve the equation x2 − 189 = 0

    Solution:

    x2 - 189 = 0 i.e. x2 = 189
    x = ±√189
    Since the value of the square root of 189 is 13.748,
    ⇒ x = +√189 or -√189 = 13.748 or -13.748.

  2. Example 2: If the surface area of a sphere is 756π in2. Find the radius of the sphere.

    Solution:

    Let 'r' be the radius of the sphere.
    ⇒ Area of the sphere = 4πr2 = 756π in2
    ⇒ r = ±√189 in
    Since radius can't be negative,
    ⇒ r = √189
    The square root of 189 is 13.748.
    ⇒ r = 13.748 in

  3. Example 3: If the area of a square is 189 in2. Find the length of the side of the square.

    Solution:

    Let 'a' be the length of the side of the square.
    ⇒ Area of the square = a2 = 189 in2
    ⇒ a = ±√189 in
    Since length can't be negative,
    ⇒ a = √189 = 13.748 in

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FAQs on the Square Root of 189

What is the Value of the Square Root of 189?

The square root of 189 is 13.74772.

Why is the Square Root of 189 an Irrational Number?

Upon prime factorizing 189 i.e. 33 × 71, 3 is in odd power. Therefore, the square root of 189 is irrational.

What is the Square Root of -189?

The square root of -189 is an imaginary number. It can be written as √-189 = √-1 × √189 = i √189 = 13.747i
where i = √-1 and it is called the imaginary unit.

What is the Value of 2 square root 189?

The square root of 189 is 13.748. Therefore, 2 √189 = 2 × 13.748 = 27.495.

What is the Square Root of 189 in Simplest Radical Form?

We need to express 189 as the product of its prime factors i.e. 189 = 3 × 3 × 3 × 7. Therefore, √189 = √3 × 3 × 3 × 7 = 3 √21. Thus, the square root of 189 in the lowest radical form is 3 √21.

Is the number 189 a Perfect Square?

The prime factorization of 189 = 33 × 71. Here, the prime factor 3 is not in the pair. Therefore, 189 is not a perfect square.

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