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# Square Root Formula

The square root formula helps in representing any number in the form of its square root. The square root of any number is that value, which when multiplied with itself gives the original number. It is represented using the "√" symbol. Every number has two square roots, one with a positive value, and the other with a negative value. For example, the number 4 has two square roots, -2 and 2. This can be expressed as √4 = ±2. This can verified as follows:** **(-2) × (-2) = 4 and 2 × 2 = 4. Let us learn more about the square root formula in this page.

## What is the Square Root Formula?

The square root of any number is given as the number raised to the power 1/2. While calculating the square root of any number, we take both the negative and the positive values as the square root after calculation. The square root formula for a perfect square would give an integer as the result. The square root of a negative number can never be a real number.

### Square Root Formula

The square root formula of a number, x is given as,

### √x = x^{1/2}

Suppose, x is any number such that, x = y × y, the formula to calculate the square root of x will be,

√x = √(y × y) = y

where, y is the square root of any number x. This also means that if the value of y is an integer, then x would be a perfect square.

## Methods for Square Root Formulas

Although there are different methods that can be conveniently used for perfect squares, the long division method can be used for any number whether it a perfect square or not.

- Repeated Subtraction Method of Square Root
- Square Root by Prime Factorization Method
- Square Root by Estimation Method
- Square Root by Long Division Method

Let us have a look at a few solved examples to understand the square root formula better.

## Examples Using Square Root Formula

**Example 1:** Using the square root formula, calculate the square root of 144.

**Solution:**

To find the square root of 144 from prime factorization of 144, we get,

144 = 2 × 2 × 2 × 2 × 3 × 3

= (2 × 2 × 3)^{2}

Using the square root formula,

√144 = ± [(2 × 2 × 3)^{2}]^{1/}^{2}

√144 = ±12

**Answer: Square root of 144 = ±12**

**Example 2: **Determine the square root of 60.

**Solution:**

To find the square root of 60 from prime factorization of 60, we get,

60 = 2 × 2 × 3 × 5

= (2)^{2 }× 3 × 5

Using the square root formula,

√60 = [(2)^{2} × 15 ]^{1/}^{2}

√60 = 2√15

**Answer: Square root of 60 = 2√15 **

**Example 3: **Calculate the length of the side of the square whose area is 400 square units.

**Solution:**

To find: Length of the side of the square.

Given, area of the square = 400 square units

Using the square root formula or more precisely the area of a square formula,

Side = √(area) = √(side)^{2}

= √400

= 20 units

**Answer: Length of the side of the square = 20 units**

## FAQs on Square Root Formula

### What is the Square Root Formula in Math?

In math, the square root formula is used to represent any number in the form of its square root, such as for any number x, its square root will be expressed as √x = x^{1/2}

### What is the Square Root Formula for Negative Numbers?

We know that the negative numbers do not have real square roots. The square roots of numbers other than a perfect square are considered irrational numbers. The principal square root of any negative number, say -x is: √(-x)= i√x, where, i is the square root of -1.

### What are the Applications of the Square Root Formula?

There are a varied number of applications of the square root formula.

- It has uses in algebra and geometry. It acts as a base for the formula of roots of a quadratic equation; quadratic fields and rings of quadratic integers.
- It is frequently used in many physical laws.
- To calculate areas, volumes, and other measurement formulas.
- Widely used by carpenters, architects, and engineers.

### How to Write the Square Root Formula in Words?

In words, the square root formula is expressed as follows. The square root of any number is the number raised to the power of 1/2.

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