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Square Root of Perfect Square Calculator

Reviewed by Calculator Editorial Team

A perfect square is an integer that is the square of another integer. For example, 16 is a perfect square because it's 4 × 4. The square root of a perfect square is simply the integer that, when multiplied by itself, gives the original number.

What is a Perfect Square?

A perfect square is an integer that can be expressed as the square of another integer. In mathematical terms, a number n is a perfect square if there exists an integer k such that n = k².

Examples of perfect squares include:

  • 0 (0²)
  • 1 (1²)
  • 4 (2²)
  • 9 (3²)
  • 16 (4²)
  • 25 (5²)
  • 36 (6²)
  • 49 (7²)
  • 64 (8²)
  • 81 (9²)

Perfect squares have several interesting mathematical properties, including:

  • They are always non-negative
  • They have an odd number of factors
  • Their square roots are integers
  • They can be represented as the sum of consecutive odd numbers

How to Find the Square Root of a Perfect Square

Finding the square root of a perfect square is straightforward because you're essentially looking for the integer that, when multiplied by itself, equals the original number.

Method 1: Using Multiplication

  1. Start with the given perfect square number
  2. Find an integer that, when multiplied by itself, equals the original number
  3. The integer you find is the square root

Formula

For a perfect square n = k², the square root is k where k is an integer.

Method 2: Using Prime Factorization

  1. Factorize the perfect square into its prime factors
  2. Group the prime factors into pairs
  3. Take one factor from each pair to find the square root

For example, to find the square root of 36:

  1. Factorize 36: 2 × 2 × 3 × 3
  2. Group the factors: (2 × 2) × (3 × 3)
  3. Take one from each group: 2 × 3 = 6

The square root of 36 is 6.

Note

This method works well for larger perfect squares but may be more complex for smaller numbers.

Examples of Square Roots of Perfect Squares

Let's look at several examples to illustrate how to find the square roots of perfect squares:

Example 1: 16

Find the square root of 16.

Solution:

  1. We need to find an integer k such that k² = 16
  2. Testing k = 4: 4 × 4 = 16
  3. Therefore, the square root of 16 is 4

Example 2: 64

Find the square root of 64.

Solution:

  1. We need to find an integer k such that k² = 64
  2. Testing k = 8: 8 × 8 = 64
  3. Therefore, the square root of 64 is 8

Example 3: 100

Find the square root of 100.

Solution:

  1. We need to find an integer k such that k² = 100
  2. Testing k = 10: 10 × 10 = 100
  3. Therefore, the square root of 100 is 10

Example 4: 144

Find the square root of 144.

Solution:

  1. We need to find an integer k such that k² = 144
  2. Testing k = 12: 12 × 12 = 144
  3. Therefore, the square root of 144 is 12

Frequently Asked Questions

What is the difference between a perfect square and a non-perfect square?

A perfect square is an integer that is the square of another integer (e.g., 16 is 4²). A non-perfect square is an integer that is not the square of another integer (e.g., 15).

How can I tell if a number is a perfect square?

You can tell if a number is a perfect square by checking if its square root is an integer. For example, √16 = 4, which is an integer, so 16 is a perfect square.

What is the square root of 0?

The square root of 0 is 0 because 0 × 0 = 0. Zero is considered a perfect square.

Can negative numbers be perfect squares?

No, perfect squares are always non-negative because a negative number multiplied by itself gives a positive result. For example, (-4) × (-4) = 16, which is a perfect square.