Square Root

Last Updated : 29 Jan, 2026

The square root of a number is essentially the value that, when multiplied by itself, yields the original number. For example square root of 9 is 3 because when we multiply 3 by 3, we get 9.

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Square ↔ Square Root

Let's take another number, 16. If you multiply 4 by itself, you get 16 (4 × 4 = 16). In this case, we say 4 is the square root of 16. The exponent for squares is 2, and for square roots, it's 1/2. So, the square root of a number n is written as √n or n1/2, where n is a positive number.

The symbol for the square root is commonly written as √, and it's known as a radical symbol.

Square Roots of First 30 Numbers

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Square Root Values (1–30)

Square Root by Prime Factorization Method

To determine the square root of a number using the prime factorization method, follow these steps:

  • Step 1: Break down the given number into its prime factors.
  • Step 2: Group the factors into pairs, ensuring that both factors in each pair are the same.
  • Step 3: Select one factor from each pair.
  • Step 4: Multiply the chosen factors together.
  • Step 5: The result of this multiplication represents the square root of the given number.

Example: Find the square root of 324 by the Prime Factorization Method.

324 = 22 × 34

Since we are finding the square root, pair the prime factors in twos: 2 × 32

Multiply the prime factors: 2 × 32 = 2 × 9 = 18

The square root of 324 is √324 = 18

So, √324 = 18 using the prime factorization method.

Square Root of Negative Number

The square root of a negative number is not a real number, because squaring any real number always gives a positive result (or zero). To handle square roots of negative values, we use complex numbers.

The principal square root of −x is written as:

\sqrt{x} = i\sqrt{x}

where i is the imaginary unit defined as:

i = \sqrt{-1}

Let's consider an example with the perfect square number 9. Now, if we look at the square root of -9, there isn't a real number solution. Expressing it using the complex number approach, √(-9) becomes √9 × √(-1), resulting in 3i {given that √(-1) = i}. In this way, 3i serves as a square root of -9.

Also check - Square Roots of Decimals

Properties of Square Root

Various Properties of Square Root are,

  • If a number is a perfect square, it has a perfect square root.
  • If a number has an even number of zeros at the end, it can have a square root.
  • You can multiply two square roots together. For example, if you multiply √3 by √7, the result is √21.
  • Multiplying two identical square roots gives a non-square root number. For instance, √4 multiplied by √4 equals 4.
  • The square root of negative numbers is not defined because perfect squares cannot be negative.
  • If a number ends with 2, 3, 7, or 8 in the unit digit, it does not have a perfect square root.
  • If a number ends with 1, 4, 5, 6, or 9 in the unit digit, it may have a perfect square root.

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