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Square Root of a Number(Meaning, Formulas & Examples)

how to find the square root of a number

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You know the area of a rectilinear figure square = $\left(\text{side} \right) \times \left(\text{side} \right)$, where $\text{side}$ is the length of edge(or side) of a square. If you want to find the length of an edge of a square whose area is known. How will you do that? In such cases, we find the square root of a number. For example, if area of a square is $256 \text{cm}^2$, then the length of its edge(or side) is $16 \text{cm}$.

Let’s understand what is a square root of a number, how to find the square root of a number, and what are the properties of the square root of a number.

What is Square Root?

The square root of a number is that factor of a number which when multiplied by itself gives the original number. Squares and square roots are special exponents. 

For example, consider the number $25$. When $5$ is multiplied by itself, it gives $25$ as the product. This can be written as $5 \times 5$ or $5^{2}$. Here, the exponent is $2$, and we call it a square. Now when the exponent is $\frac{1}{2}$, it is called as the square root of the number. For example, $\sqrt{n}=n^\frac{1}{2}$, where $n$ is a positive integer.

The square root of a number is the inverse operation of squaring a number. The square of a number is the value that is obtained when we multiply the number by itself, while the square root of a number is obtained by finding a number that when squared gives the original number.

If square of a number $a$ is $b$, then square root of the number $b$ is $a$.

girl-with-teacher-happy
Maths can be really interesting for kids

How to Find the Square Root of a Number?

It is very easy to find the square root of a number that is a perfect square. Perfect squares are those positive numbers that 

can be expressed as the product of a number by itself. in other words, perfect squares are numbers that are expressed as the value of power $2$ of any integer. For example, $9$ is a perfect square number, since $3^{2} = 9$. We can use any of these four methods to find the square root of numbers.

  • Repeated Subtraction Method of Square Root
  • Square Root by Prime Factorization Method
  • Square Root by Estimation Method

Repeated Subtraction Method of Square Root

This is a very simple method. In this case, we use one of the properties of square numbers. The property states that the “Sum of first $n$ odd numbers is equal to $n^{2}$”

We subtract the consecutive odd numbers from the number for which we are finding the square root, till we reach 0. The number of times we subtract is the square root of the given number. This method works only for perfect square numbers. Let us find the square root of $36$ using this method.

$36 – 1 = 35$

$35 – 3 = 32$

$32 – 5 = 27$

$27 – 7 = 20$

$20 – 9 = 11$

$11 – 11 = 0$

You can observe that we have subtracted $6$ times. Thus, $\sqrt{36} = 6$.

Square Root by Prime Factorization Method

To find the square root of a given number through the prime factorization method, we follow the steps given below:

Step 1: Divide the given number into its prime factors.

Step 2: Form pairs of similar factors such that both factors in each pair are equal.

Step 3: Take one factor from the pair.

Step 4: Find the product of the factors obtained by taking one factor from each pair.

Step 5: That product is the square root of the given number.

Let’s consider some examples to understand the process.

Examples

Let’s consider some examples to understand the process.

Ex 1: Find the square root of $144$.

Prime factorization of $144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 = 2^{4} \times 3^{2} = \left(2^{2} \right)^{2} \times 3^{2} = 4^{2} \times 3^{2} = (4 \times 3)^{2} = 12^{2}$.

Thus, S\sqrt{144} = 12$.

Ex 2: Find the square root of $324$.

Prime factorization of $324 = 2 \times 2 \times 3 \times 3 \times 3 \times 3 = 2^{2} \times 3^{4} = 2^{2} \times (3{2})^{2} = 2^{2} \times 9^{2} = \left(2 \times 9 \right)^{2} = 18^{2}$.

Thus, $\sqrt{324} = 18$.

Square Root by Estimation Method

Estimation and approximation refer to a reasonable guess of the actual value to make calculations easier and more realistic. This method helps in estimating and approximating the square root of a given number. This method is used for numbers that are not perfect squares. 

Let’s consider some examples to understand the process.

Examples

Ex 1: Find $\sqrt{15}$. 

First, find the nearest perfect square numbers to which are less than and greater than $15$. 

These numbers are 

  • $9$ on the lower side since, $3 \times 3 = 9$
  • $16$ on the upper side since, $4 \times 4 = 16$

This means that $\sqrt{15}$ lies between $3$ and $4$. 

Now, we need to see if $\sqrt{15}$ is closer to $3$ or $4$. 

Since, $3^{2} = 9$ and $4^{2} = 16$, therefore, $\sqrt{15}$ lies between $3.5$ and $4$ and is closer to $4$.

Let us find the squares of numbers $3.8$ and $3.9$ (numbers closer to $4$). 

Since $3.8^{2} = 14.44$ and $3.9^{2} = 15.21$. This implies that $\sqrt{15}$ lies between $3.8$ and $3.9$. 

Now, repeat the process and check between $3.85$ and $3.9$. 

Proceeding in this way, we can observe that $\sqrt{15} = 3.872$.

Note: 

  • The number of times the process is repeated depends on the number of decimal places required.
  • This is a very long process and time-consuming.

Square Root Formula

The square root of a number has the exponent of $\frac{1}{2}$. The square root formula is used to find the square root of a number. We know the exponent formula: $\sqrt[n]{x} = x^{\frac{1}{n}}$.

When $n= 2$, we call it square root. We can use any of the above methods for finding the square root, such as prime factorization, long division, and so on. 

For example, $16^{\frac{1}{2}} =\sqrt{16} = \sqrt{4 \times 4} = 3$. So, the formula for writing the square root of a number is $\sqrt{x} = x^{\frac{1}{2}}$.

Square Root of a Negative Number

The square root of a negative number cannot be a real number, since a square is either a positive number or zero. But complex numbers have the solutions to the square root of a negative number. 

The principal square root of $-x$ is: $\sqrt{-x} = i\sqrt{x}$. Here, $i$ is the square root of $-1$ and is called an imaginary unit.

Let’s consider some examples to understand the process.

Examples

Ex 1: Square root of $-16$.

$-16 = 16 \times \left(-1 \right)$

$\sqrt{-16} \sqrt{16 \times \left(-1 \right)} = \sqrt{16} \times i = 4i$.

Ex 2: Square root of $-\frac{25}{36}$.

$-\frac{25}{36} = \frac{25}{36} \times \left(-1 \right)$

Therefore, $\sqrt{-\frac{25}{36}} = \sqrt{-1 \times \frac{25}{36}} = \sqrt{-1} \times \sqrt{\frac{25}{36}} = i \times \frac{\sqrt{25}}{\sqrt{36}} = i \times \frac{5}{6} = \frac{5}{6}i$.

Conclusion

The square root of a number is that factor of a number which when multiplied by itself gives the original number. There are $3$ methods of finding the square root of a number – Repeated Subtraction Method of Square Root, Square Root by Prime Factorization Method, and Square Root by Estimation Method.

Practice Problems

  1. Find the square root of the following numbers using the repeated subtraction method
    • $25$
    • $121$
    • $196$
  2. Find the square root of the following numbers using the prime factorization method
    • $196$
    • $1296$
    • $784$
  3. Find the square root of the following numbers using the approximation method
    • $31$
    • $53$
    • $90$

Recommended Reading

FAQs

What is the square root of a number?

The square root of a number is a number that when multiplied by itself gives the actual number. For example, $4$ is the square root of $16$, and this is expressed as $\sqrt{16} = 4$. This means when $4$ is multiplied by $4$ it results in $16$.

How to find the square root of a number?

It is very easy to find the square root of a number that is a perfect square. For example, $25$ is a perfect square, $25 = 5 \times 5$. So, $5$ is the square root of $25$ and this can be expressed as $\sqrt{25} = 5$. The square root of any number, in general, can be found by using any of the three methods given below:
a) Repeated Subtraction Method
b) Prime Factorization Method
c) Estimation or Approximation Method

Can square root be negative?

Yes, the square root of a number can be negative. In fact, all the perfect squares like $4$, $9$, $25$, etc. have two square roots, one is a positive value and one is a negative value. 

For example, the square roots of $4$ are $-2$ and $2$. Since, $\left(-2 \right) \times \left(-2 \right)$ is also equal to $4$. Similarly, the square roots of $9$ are $3$ and $-3$, and so on.

What is the formula for calculating the square root of a number?

The square root of any number can be expressed using the formula: $\sqrt{x} = x^{\frac{1}{2}}$. In other words, if a number has $\frac{1}{2}$ as its exponent, it means we need to find the square root of the number.

What are the applications of the square root formula?

There are various applications of the square root formula:
a) The square root formula is mainly used in algebra and geometry.
b) It helps in finding the roots of a quadratic equation.
c) It is widely used by engineers.

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