Square Root – Formula, Examples | How to Find Square Root?

The square root of a number is the value that, when multiplied by itself, results in the original number. It is a special exponent related to squares. Take the number 16 as an example. When 4 is multiplied by itself, the product is 16, expressed as 4 × 4 or 3². In this case, the exponent is 2, denoting a square. Now, if the exponent is 1/2, it signifies the square root of the number. For instance, √n is equivalent to n^(1/2), where ‘n’ represents a positive integer.

What is a Square Root?

A square root of a number ‘a’ is another number ‘b’ such that when ‘b’ is squared, the result is ‘a’. Symbolically, if b² = a, then b is the square root of a.

Square Root Symbol

The symbol used to represent the square root in mathematics is the radical symbol (√). The radical symbol is placed before the number or expression for which the square root is to be calculated. The expression written under the radical symbol indicates the radicand, the number for which the square root is being taken.

For example:

The square root of 25 is written as √25 ​, where √ is the square root symbol, and 25 is the radicand.

Formula for Square Root:

The square root of a number ‘a’ is denoted by √a. Mathematically, \(b^{2}\) =a, then b is the square root of a. The general formula for the square root is:

\(\sqrt{a} = \pm \sqrt{b} \)

Properties of Square root

1. Non-Negative Result: The square root of a non-negative real number is always non-negative. For instance, √9 equals 3, not -3.
2. Multiplying Square Roots: The square root of a product is the same as the product of the square roots. Symbolically, √(ab) = √a * √b.
3. Dividing Square Roots: The square root of a quotient is the same as the quotient of the square roots. Symbolically, \(\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}} \)

How do Find Square Root of Numbers?

Finding the square root of a number involves determining a value that, when multiplied by itself, results in the original number. Here are several methods to find the square root of numbers:

1. Prime Factorization Method:

• Express the number as a product of its prime factors.
• Group the factors into pairs.
• The square root is the product of one factor from each pair.

2. Estimation:

• For non-perfect squares, use estimation to find a close approximation of the square root.

4. Special Square Roots:

• Memorize the square roots of perfect squares for quick calculations.

5. Long Division Method:

• Similar to long division, this method is useful for non-perfect squares

Finding the square root of a number involves determining a value that, when multiplied by itself, results in the original number. Here are several methods to find the square root of numbers:

Example: √30

• Continue the process for desired accuracy.

These methods provide various approaches to find square roots, and the choice of method depends on the context and the nature of the number involved.

Square Root of Perfect squares

Here are some examples of the square roots of perfect squares:

Square Root of 1:

√1 = 1

Explanation: 1×1=1

Square Root of 4:

√4 = 2

Explanation: 2×2=4

Square Root of 9:

√9 = 3

Explanation: 3×3=9

Square Root of 16:

√16 = 4

Explanation: 4×4=16

Square Root of 25:

√25 =5

Explanation: 5×5=25

Square Root of 36:

√36 = 6

Explanation: 6×6=36

Square Root of 49:

√49 = 7

Explanation: 7×7=49

Square Root of Negative Number

In the realm of real numbers, the square root of a negative number is undefined. This is because, for any real number, squaring it (multiplying it by itself) results in a non-negative value. In other words, the square of a real number is always either zero or a positive number.

However, in the field of complex numbers, we introduce the concept of imaginary numbers to handle square roots of negative numbers. The imaginary unit, denoted by ‘i’, is defined as the square root of -1:

i² = -1

Using this imaginary unit, we can represent the square root of any negative real number as a combination of a real part and an imaginary part. For example:

√-9 = 3i

Here, the square root of -9 is expressed as 3 times the imaginary unit ‘i’. In general, the square root of a negative number ‘a’ is written as:

√-a = √a . i

Solved Examples on Square Roots

Example 1: What is:

1. The square root of 2
2. The square root of 4
3. The square root of 3
4. The square root of 5

Solution:

1. value of root 2 i.e. √2 = 1.4142
2. value of root 4 i.e. √4 = 2
3. value of root 3 i.e. √3 = 1.7321
4. value of root 5 i.e. √5 = 2.2361

FAQs on Square root

1. What is a square root?

• A square root of a number is a value that, when multiplied by itself, gives the original number. For a given number ‘a,’ the square root is denoted as √a.

2. How is the square root represented?

• The square root is symbolically represented by the radical symbol (√). For example, the square root of 25 is written as √25, which equals 5.

3. What does the square root of a perfect square yield?

The square root of a perfect square results in a whole number. For instance, the square root of 16 is 4, as 4² = 16

5. How do you find the square root of a number?

• The square root of a number can be found using methods like prime factorization, estimation, using a calculator, or through special square root values for perfect squares.

6. What is the square root of 0?

The square root of 0 is 0, as 0² = 0