In this chapter we will learn to find square root using prime factorization method.

At the end some solved examples are provided for better clarity.

To understand the chapter, you should have basic understanding of **square root** and prime factorization method. Click the red link to learn about the same.

## Finding Square root using prime factorization

**Follow the below steps** to find square root of given perfect square;

(a) **Do the prime factorization** of given number.

(b)** Arrange all the factors in pair of two’s**.

Here for any factor, the pair of two can be shown by using exponent 2 above the factor.

Now **there are two possibilities**;

(i)** If factors are not forming pair of two then it is not a perfect square**.

(ii) **If all factors have exponent 2, then it is a perfect square.**

Assuming that the given number is a perfect square then move to next step.

(c) **Remove exponent 2 from the factors and multiply them**.

The final result after multiplication is the square root of perfect square.

**Note:** Using prime factorization, we can find square root of only perfect square number.

### Can we calculate the square root of negative number ?

NO !!!

The square root of negative number is not a real number.

In short, the **square root of negative number doesn’t exist**.

However, there is an imaginary number system which help us define the negative square root.

The imaginary number is in the syllabus of higher grade mathematics.

## Finding Square root using Prime Factorization – Solved examples

Example 01

Find square root of 64

**Solution**

Follow the below steps;**(a) Do prime factorization.**

Note that all the factors are forming pair of 2. This means that the given number is a perfect square.

**(b) Arranging factors using exponents.**

The above prime factorization can be expressed as;

\mathtt{64=\ 2^{2} \times \ 2^{2} \times \ 2^{2} \ }

The power 2 in above factors express that they are forming pair of two’s.**(c) To get the square root, remove all the exponents and multiply the numbers.**

\mathtt{\Longrightarrow \ \sqrt{64}}\\\ \\ \mathtt{\Longrightarrow \ 2\times 2\times \ 2}\\\ \\ \mathtt{\Longrightarrow \ 8}

Hence, **square root of number 64 is 8**.

**Example 02**

Find square root of 81.

**Solution**

Follow the below steps.**(a) Do prime factorization.**

Note that number 81 is forming pair of two’s. Hence it is a perfect square.

**(b) Arrange factors using exponents.**

The factors can be arranged in pair of two’s using power 2.

\mathtt{81\ =\ 3^{2} \times 3^{2}} **(c) Calculate square root.**

To get the square root, remove all the exponents and multiply.

\mathtt{\Longrightarrow \ \sqrt{81}}\\\ \\ \mathtt{\Longrightarrow \ 3\times 3}\\\ \\ \mathtt{\Longrightarrow \ 9}

Hence, **9 is the square root of number 81.**

**Example 03**

Find the square root of 196 using prime factorization.**Solution**

To get the square root, follow the below steps;**(a) Do prime factorization.**

All the numbers are forming pair of two’s, hence the given number is a perfect square.

**(b) Arrange the numbers in pairs using exponents**

\mathtt{196\ =\ 2^{2} \times 7^{2}} **(c) Calculate the square root**

Remove all the exponents and multiply the numbers.

\mathtt{\Longrightarrow \ \sqrt{196}}\\\ \\ \mathtt{\Longrightarrow \ 2\times 7}\\\ \\ \mathtt{\Longrightarrow \ 14}

Hence, **14 is the square root of number 196**.

**Example 04**

Find the square root of number 300.**Solution**

Follow the below steps;**(a) Do prime factorization.**

All the factors are in pair of two’s. Hence, the number 400 is a perfect square.

**(b) Arrange the numbers in pairs using exponents**.

\mathtt{400\ =\ 2^{2} \times 2^{2} \times 5^{2}}

**(c) Calculate the square root**

To get the square root, remove all the exponents and multiply the numbers.

\mathtt{\Longrightarrow \ \sqrt{400}}\\\ \\ \mathtt{\Longrightarrow \ 2\ \times \ 2\times \ 5}\\\ \\ \mathtt{\Longrightarrow \ 20}

Hence, **number 20 is the solution.**

**Example 05**

Calculate square root of 80.**Solution****(a) Do prime factorization.**

Note that **factor 5 do not have any pair**. So,** number 80 is not a perfect square**.

The above prime factorization can be expressed as;

\mathtt{80\ =\ 2^{2} \times 2^{2} \times 5}

Since number 80 is not a perfect square, **we cannot calculate exact square root value using prime factorization method**.

However, we can express the square root as;

\mathtt{\Longrightarrow \ \sqrt{80}}\\\ \\ \mathtt{\Longrightarrow \ \sqrt{\mathtt{\ 2^{2} \times 2^{2} \times 5}}}\\\ \\ \mathtt{\Longrightarrow \ 2\ \times \ 2\times \ \sqrt{5}}\\\ \\ \mathtt{\Longrightarrow \ 4\sqrt{5}}

Hence, \mathtt{4\sqrt{5}} is the square root of given number.

**Example 06**

Find square root of 484

**Solution**

Follow the below steps;**(a) Do prime factorization**.

Note that all the factors are forming pair of two’s, hence the given number is a perfect square.**(b) Arrange the numbers in pairs using exponents**.

\mathtt{484\ =\ 2^{2} \times 11{^{2}}} **(c) Calculate the square root.**

Remove all the exponents and multiply the numbers.

\mathtt{\Longrightarrow \sqrt{484}}\\\ \\ \mathtt{\Longrightarrow \ \sqrt{\mathtt{\ 2^{2} \times 11{^{2}}}}}\\\ \\ \mathtt{\Longrightarrow \ 2\ \times \ 11}\\\ \\ \mathtt{\Longrightarrow \ 22}

Hence, **22 is the square root of number 484.**

**Example 07**

Find square root of 84.**Solution****(a) Do prime factorization**

Note that the **factors 3 & 7 do not have any pairs**. Hence, the **number 84 is not a perfect square**.

**(b) Arrange the numbers using exponents**.

\mathtt{84\ =\ 2^{2} \times 3\times 7}

Since the number is not a perfect square, we cannot calculate the exact value of square root.

However, we can express the square root as;

\mathtt{\Longrightarrow \sqrt{84}}\\\ \\ \mathtt{\Longrightarrow \ \sqrt{\mathtt{\ 2^{2} \times 3\times 7}}}\\\ \\ \mathtt{\Longrightarrow \ 2\sqrt{3\times 7}}\\\ \\ \mathtt{\Longrightarrow \ 2\sqrt{21}}

Hence, \mathtt{2\sqrt{21}} is the square root of number 84.

**Example 08**

Find the square root of 1764.

**Solution**

Follow the below steps;**(a) Do Prime factorization**.

Note that all the factors are forming pair of two, so the given number is perfect square.**(b) Arrange the pairs using exponents.**

\mathtt{1764\ =\ 2^{2} \times 3^{2} \times 7^{2}} **(c) Calculate the square root.**

Remove the exponents and multiply the numbers.

\mathtt{\Longrightarrow \sqrt{1764}}\\\ \\ \mathtt{\Longrightarrow \ \sqrt{\mathtt{\ \ 2^{2} \times 3^{2} \times 7{^{2}}}}}\\\ \\ \mathtt{\Longrightarrow \ 2\times 3\times 7}\\\ \\ \mathtt{\Longrightarrow \ 42}

Hence,

**42 is the square root of number 1764.**