Finding Nth roots of any number is an essential mathematical skill that is often tested in competitive exams such as the SAT, GRE, GMAT, and CAT frequently. This is because the ability to calculate roots is a fundamental skill in mathematics, and it is applicable in many real-world situations, including finance, engineering, and science.

Knowing how to find the nth root of any number quickly and accurately can save time in competitive exams and increase one’s overall score. In this chapter, we are going to discuss how to find the square root, cube root, and even the Nth root of a number in a time frame of a few seconds.

**Division method of finding the square root**

This is a traditional way of solving square-root problems. Here we are going to find the square root of 51573.

**Step 1: **Make a two-two-digit pair from the right side.

**Step 2: **Here the product of 2 (to the left of 5) and 2 (below the first 2) is equal to 4 which is less than 5. So write the number 2 three times as shown in the picture (one on the left, one below the first 2, and the third on top)

**Step 3: **Write the second pair i.e. 15 with remainder 1. Now, after shifting the 4 to the tens place, repeat the same process as before.

42×2=84

84<115

Remainder= 31

**Step 4:** Repeat the process done in step:3

447×7=3129

3129<3173

Remainder= 44

**Step 5:** Put a decimal point after 227 and add 00 to the right of remainder 44 to make the new number 4400.

**Step 6: **Repeat the process done in step:4

4540×0=0

0<4400

Remainder= 4400

**Step 7: **Repeat the process done in step:6

45409×9=408681

408681<440000

End this process now.

The square root of 51573 up to two decimal places= 227.09

**Que: Find the square root of 4586 up to two decimal places.**

**Solution:** \sqrt{4586}=67.72 (Approx.)

**The square root of a perfect square**

Here are concepts and tricks to find the square root of any perfect square within a few seconds without pen and paper.

**Perfect Square:** A perfect square is a non-negative integer that is the square of another integer.

Examples: 4, 9, 16, 25, etc.

Number^{2} | Perfect Square |

****0^{2} | *******0 |

****1^{2} | *******1 |

****2^{2} | *******4 |

****3^{2} | *******9 |

****4^{2} | *******6 |

****5^{2} | *******5 |

****6^{2} | *******6 |

****7^{2} | *******9 |

****8^{2} | *******4 |

****9^{2} | *******1 |

This means that if the unit digit of a perfect square is given, then we can easily find the unit digit of its square root.

Unit digit of a Perfect Square | Unit digit of the possible Square Root |

0 | 0 |

1 | 1 or 9 |

4 | 2 or 8 |

5 | 5 |

6 | 4 or 6 |

9 | 3 or 7 |

Note: 2,3,7, and 8 cannot be the unit digit of a perfect square number.

**Que 1: Find the square root of 7921.**

Note: This method is only applicable to perfect square numbers.

**Step 1: **Write down the last two digits of the number and remove it from the original number.

Last two digits ⇒21

New number ⇒79

**Step 2: **Check its unit digit and accordingly find the unit digit of the possible square root.

Unit Digit ⇒1

So, the unit digit of possible square root ⇒1 or 9 ………….(i)

**Step 3:** Find the value of N where the square of N is less than or equal to the new number.

N^{2 }≤79

N=8 ………(ii)

**Step 4: **The real square root is either 81 or 89

**Step 5: **If New Number < N×(N+1)⇒ the smaller number is correct

If New Number ≥ N×(N+1)⇒ the large number is correct

N×(N+1)=8×9=72

79≥ 72

So, the correct square root is 89

**Que 2: Find the square root of 26244.**

Step 1: 262~~44~~

Step 2: The unit digit of possible square root ⇒2 or 8

Step 3: N^{2 }≤262

N=16

Step 4: The real square root is either 162 or 168

Step 5: N×(N+1)=16×17=272

New number<N×(N+1)

So, the correct square root is 162.

**Que 3: Find the square root of 60516.**

Step 1: 605~~16~~

Step 2: The unit digit of possible square root ⇒4 or 6

Step 3: N^{2 }≤605

N=24

Step 4: The real square root is either 244 or 246

Step 5: N×(N+1)=24×25=600

New number ≥ N×(N+1)

So, the correct square root is 246.

**The Cube Root of a Perfect Cube**

A perfect cube is an integer that is the cube of another integer.

Examples: 1, 8, 27, 64, etc.

Number^{3} | Perfect Cube |

****0^{3} | *******0 |

****1^{3} | *******1 |

****2^{3} | *******8 |

****3^{3} | *******7 |

****4^{3} | *******4 |

****5^{3} | *******5 |

****6^{3} | *******6 |

****7^{3} | *******3 |

****8^{3} | *******2 |

****9^{3} | *******9 |

This means that if the unit digit of a perfect cube is given, then we can easily find the unit digit of its cube root.

**Que 1: Find the cube root of 6859.**

**Step 1: **Write down the last three digits of the number and remove it from the original number.

Last three digits ⇒859

New number ⇒6

**Step 2: **Check its unit digit and accordingly find the unit digit of the possible cube root.

Unit Digit ⇒9

So, the unit digit of possible cube root ⇒9 ………….(i)

**Step 3:** Find the value of N where the cube of N is less than or equal to the new number.

N^{3 }≤6

N=1 ………(ii)

**Step 4: **The correct cube root is 19.

**Que 2: Find the cube root of 592704.**

Step 1: 592~~704~~

Step 2: The unit digit of possible cube root ⇒4

Step 3: N^{3 }≤592

N=8

Step 4: The correct cube root is 84.

**The square of any number**

Here we would use a very common formula of the square but in a different way

**(a+b)²=a²+2ab+b²**

**Que 1: Find the value of 43²**

Step 1: Break the number into two parts (a & b)

where ‘b’ is the unit digit and ‘a’ is the remaining part.

Step 2: a² / 2ab / b²

=4² / 2×4×3 / 3²

=16 / 24 / 9

=16+2 / 4 / 9

=18 / 4 / 9

=1849

The required result=1849

**Que 2: Find the value of 127²**

Step 1: Break the number into two parts (a & b).

where ‘b’ is the unit digit and ‘a’ is the remaining part.

Step 2: a² / 2ab / b²

=12² / 2×12×7 / 7²

=144 / 168 / 49

=144 / 168+4=172 / 9

=144 / 172 / 9

=144+17 / 2 / 9

=161 / 2 / 9

The required result=16129

**The cube of any number**

Here we would use a very common formula of the cube but in a different way

**(a+b)**^{3}**=a²+3a**^{2}**b+3ab**^{2}**+b**^{3}

**Que 1: Find the value of 43**^{3}

Solution: 4^{3}/ 3×4^{2}×3/ 3×4×3^{2}/ 3^{3}

=64/ 144/ 108/ 27

=64/ 144/ 108+2=110/ 7

=64/ 144+11=155/ 0/ 7

=64+15=79/ 5/ 0/ 7

=79507

**Que 2: Find the value of 127**^{3}

Solution: 12^{3}/ 3×12^{2}×7/ 3×12×7^{2}/ 7^{3}

=1728/ 3024/ 1764/ 343

=1728/ 3024/ 1764+34=1798/ 3

=1728/ 3024+179=3203/ 8/ 3

=1728+320=2048/ 3/ 8/ 3

=2048383

**Que 3: Find the value of 21**^{3}

Solution: 2^{3}/ 3×2^{2}×1/ 3×2×1^{2}/ 1^{3}

=8/ 12/ 6/ 1

=9261

**The nth root of any number**

Find the concepts and tricks to find the nth root of any number in the given link.

All concepts and tricks of Simplification and Approximation: Click Here

The Simplification and Approximation chapter covers some very important concepts which can increase your calculation speed to a very high level.

- Concept of Remainders
- Remainder Theorem
- Concept of Negative remainder, devisor, or dividend
- Divisibility Rules
- Unit Digits
- Last two and Last three digits
- Mixed Fractions
- Simplification
- Approximation
- Use of percentages to reduce the complexity of complicated calculations
- Digital Sum/ Digital Root, etc.

Geometry Basics: Click Here