**Introduction to Square Root**

We have learnt about exponents.

Squares and square roots are special exponents.

Consider the number \(9\)

Which number when multiplied by itself gives \(9\) as the product?

When the exponent is \(2\), it is called a square.

When the exponent is \(\frac{1}{2}\), it is called a square root.

**How to Find Square Root**

We can use the subtraction method, prime factorisation method, approximation method and long division method to find the square root of a given number.

**Repeated Subtraction Method **

This is a very simple method.

We will 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 \(16\) using this method.

\[\begin{align}

16 -1 &= 15 \\

15 - 3 &=12 \\

12 - 5 &= 7 \\

7 - 7 &= 0

\end{align}\]

You can observe that we have subtracted \(4\) times.

Thus, \(\sqrt{16} = 4\)

**Prime Factorisation**

To find the square root of a given number through the prime factorization method:

- Divide the given number into its prime factors.
- Form pairs of similar factors such that both factors in each pair are equal.
- Take one factor from the pair.
- Find the product of the factors obtained by taking one factor from each pair.
- That product is the square root of the given number.

Let us find the square root of 144 by this method.

This method works when the given number is a perfect square number.

**Estimation and Approximation**

This method helps in estimating and approximating the square root of a given number.

Let us use this method to find the \(\sqrt {15}\)

Find the nearest perfect square numbers to \(15\)

\(9\) and \(16\) are the perfect square numbers nearest to \(15\)

We know that \(\sqrt{16}\) = \(4\) and \(\sqrt{9}\) = \(3\)

This implies that \(\sqrt {15}\) lies between \(3\) and \(4\)

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

Let us consider \(3.5\) and \(4\)

\(3.5^2 = 12.25\) and \(4^2 = 16\)

Thus, \(\sqrt {15}\) lies between \(3.5\) and \(4\) and is closer to \(4\)

Let us find the squares of \(3.8\) and \(3.9\)

\(3.8^2 = 12.25\) and \(3.9^2 = 15.21\)

This implies that \(sqrt {15}\) lies between \(3.8\) and \(3.9\)

We can repeat the process and check between \(3.85\) and \(3.9\)

We can observe that \(\sqrt{15}\) = \(3.872\)

This is a very long process and time-consuming.

**Long Division Method **

We can find the exact square root of any given number using this method.

Let us understand this process with an example.

Let's find the square root of \(180\)

**Step 1:** Place a bar over every pair of digits of the number starting from the unit’s place (right-most side).

We will have two pairs, i.e. \(1\) and \(80\)

**Step 2:** We divide the left-most number by the largest number whose square is less than or equal to the number in the left-most pair.

**Step 3:** Bring down the number under the next bar to the right of the remainder.

Add the last digit of the quotient to the divisor.

To the right of the obtained sum, find a suitable number which together with the result of the sum forms a new divisor for the new dividend that is carried down.

**Step 4:** The new number in the quotient will have the same number as selected in the divisor.

The condition is the same as being either less or equal to that of the dividend.

**Step 5:** Now, we will continue this process further using a decimal point and adding zeros in pairs to the remainder.

**Step 6:** The quotient thus obtained will be the square root of the number.

**Square Root Table**

Here are the square root list of perfect square numbers and some non-perfect square numbers from \(1\) to \(10\)

Number | Square root |
---|---|

\(1\) | \(1\) |

\(2\) | \(1.414\) |

\(3\) | \(1.732\) |

\(4\) | \(2\) |

\(5\) | \(2.236\) |

\(6\) | \(2.449\) |

\(7\) | \(2.646\) |

\(8\) | \(2.828\) |

\(9\) | \(3\) |

\(10\) | \(3.162\) |

Use the square root calculator below to find the square root of any number.

The square roots of numbers that are not a perfect square are part of irrational numbers.

For quick calculations, it is good to memorise the square root list shown above.

**Square Root Calculator**

Enter the number for which you need to calculate the square root.

**Square Root Symbol and Formula**

Square root is nothing but the exponent \(\frac{1}{2}\)

**Square Root Symbol**

It is represented by the radical symbol pronounced as square root.

Example: \(\sqrt{9}\) = \(3\) as (\(3 \times 3 = 9\))

**Square Root Formula**

We know the exponent formula

\( \sqrt[\text{n}]{x } = x ^\frac{1}{n}\)

When \(n\) = \(2\), we call it square root.

We can use any of the above methods for finding the square root like prime factorisation, long division and so on.

- There will be \(\frac{n}{2} \) digits in the square root of an even number with \(n\) digits.

There will be \(\frac{n+1}{2} \) digits in the square root of an odd number with \(n\) digits. - Square root of a number ending with:

\(1\) will end in \(1\) or \(9\)

\(4\) will end in \(2\) or \(8\)

\(9\) will end in \(3\) or \(7\)

\(6\) will end in \(4\) or \(6\)

**Solved Examples on Square Root **

Example 1 |

Find the square root of 169 by repeated subtraction method.

**Solution: **

\[\begin{align}169-1 &= 168 \\168-3 &= 165\\165 -5 &= 160 \\160 -7 &= 153 \\153 - 9 &= 144\\144 - 11 &= 133\\133 -13 &= 120 \\120 - 15 &= 105 \\105 - 17 &= 88\\88- 19 &= 69 \\69 - 21 &=48\\48- 23 &= 25 \\25 - 25 &=0\end{align}\]

We have done repeated subtraction \(13\) times.

\(\therefore \sqrt{169} = 13\) |

Example 2 |

Find the square root of \(529\) by prime factorisation method.

**Solution: **

Prime factorisation of \(529\)

We can see that,

\[\begin{align}

529 &= 23 \times 23 \\

\Rightarrow\sqrt{529} &= 23

\end{align}\]** **

\(\therefore \sqrt{529} = 23\) |

Example 3 |

Find the square root of \(6\) to \(3\) decimal places by long division method.

**Solution: **

\(\therefore \sqrt{6} = 2.449\) |

Example 4 |

Estimate the square root of \(7\) to \(2\) decimal places using the approximation method.

**Solution: **

The nearest square numbers to \(7\) are \(4\) and \(9\)

Therefore, \(\sqrt{7}\) lies between \(\sqrt{4}\) and \(\sqrt{9}\) which are \(2\) and \(3\)

We know that,

\(2.5^2 = 6.25\) , \( 2.6^2 = 6.76\) and \(2.7^2 = 7.29\)

So \(\sqrt{7}\) lies betwen \(2.6\) and \(2.7\)

Let us find the square of \(2.65\)

\(2.65^2 = 7.0225 \)

\(\therefore \sqrt{7} = 2.65 \) |

Example 5 |

John was baking a square cake for his friend's birthday party.

He decided to give each guest a square piece of cake of sides \(1\:\text{inch}\).

There are \(121\) guests attending the party.

What should be the length of the sides of his square cake tin so that he can serve all the guests \(1\) piece each?

**Solution: **

Number of pieces of size \(1\:\text{inch}\) by \(1\:\text{inch}\) =

\(121\)

This is the area of the square cake tin.

Therefore,

Length of the cake tin =

\[\begin{align}\sqrt{121} = 11\end{align}\]

Therefore, the cake tin size choosen by John is:

\(11\:\text{inch}\) by \(11\:\text{inch}\) |

**Practice Questions**

**Here are few problems for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.**

- Evaluate \( \sqrt{\frac{169}{25}} - \sqrt{\frac{64}{49}} +\sqrt{\frac{196}{9}}\)
- For what value of \(x\) is this statement valid?

\( \frac{\sqrt{128}}{x} = \frac{x}{\sqrt{162}} \)

**Maths Olympiad Sample Papers**

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

- IMO Sample Paper Class 1
- IMO Sample Paper Class 2
- IMO Sample Paper Class 3
- IMO Sample Paper Class 4
- IMO Sample Paper Class 5
- IMO Sample Paper Class 6
- IMO Sample Paper Class 7
- IMO Sample Paper Class 8
- IMO Sample Paper Class 9
- IMO Sample Paper Class 10

To know more about the Maths Olympiad you can **click here**

**Frequently Asked Questions(FAQs)**

## 1. How do I calculate square root?

- To calculate the square root of perfect square numbers, we can use:

Prime factorisation

Repeated subtraction method - To calculate the square root of a non-perfect square number, we can use:

Approximation method

Long division method

## 2. How do you simplify square roots?

To simplify a square root, we need to find the prime factorisation of the given number.

If a factor cannot be grouped, retain them under the square root symbol.

Example: \(\sqrt{12} = \sqrt{2 \times 2\times3} = 2\sqrt{3}\)

## 3. What is the square root of 80?

Square root of \(80\) in the decimal form is \( {\displaystyle \pm}8.9442 \) rounded to \(4\) decimal places.

Use the calculator above to find the square root of any given number

\(\sqrt{80} \!= \!\sqrt{5 \times2 \times 2 \!\times\! 2\times2}\! =\!4\sqrt{5}\)

## 4. What is the square root of 4?

The square root of 4 is \({\displaystyle \pm}2\)

( \(2 \times 2 = 4\)) and

(\(-2 \times -2 = 4 \))

## 5. What are the real square roots of 100?

The real square roots of 100 are \({\displaystyle \pm}10\)

( \(10 \times 10 = 100\)) and

(\(-10 \times -10 = 100 \))