In this post we will learn how to solve problems based on exponents i.e A^{B}.In exams generally questions based on finding the last digit in expression A^{B} are asked. So let us start learning this topic. Must learn topic for CAT,SSC,Banking and various other competitive exams.**FINDING LAST DIGIT IN A ^{B} :**

Let us take an example for better understanding.

Q. Find the last digit in 2347

^{41}?

A. As from sight this question appears to be very typical but if we go by the rules it is very easy to solve such type of problems. In this question we have to find the last digit in 2347

^{41}for solving this type of problems we will only concentrate on the unit digit of the Number i.e in this case 7 and will find the last digit of 7

^{41}not for the whole Number 2347

^{41}

Next part of solving this problem is based on Cyclicity of the number.Let us learn about cyclicity of a number.

**a**, units place digit of the result depends on units place digit of

^{b}*and the divisibility of power*

**a****.**

*b*Consider powers of 2

As we know,

^{1}=2

2

^{2}= 4

2

^{3}= 8

2

^{4}= 16

2

^{5}= 32

2

^{6}= 64

2

^{7}= 128.. and so on

*What do you observe here?*We can see that the units place digit for powers of 2 repeat in an order: 2, 4, 8, 6. So the "cyclicity" of number 2 is 4 (

*that means the pattern repeats after 4 occurrences*) and the cycle pattern is 2, 4, 8, 6. From this you can see that to find the units place digit of powers of 2, you have to divide the exponent by 4.

Let's check the validity of above formula with an example.

**Example**

- Find the units place digit of 2
^{99}?

Using the above observation of cyclicity of powers of 2, divide the exponent by 4. 99/4 gives reminder as 3. That means, units place digit of 2

^{99}is the 3rd item in the cycle which is 8.

**Shortcuts to solve problems related to units place digit of a**^{b}**Case 1: If b is a multiple of 4**- If
**a**is an even number, ie: 2, 4, 6 or 8 then the units place digit is 6 - If
**a**is an odd number, ie: 1, 3, 7 or 9 then the units place digit is 1

- If
**Case 2: If b is not a multiple of 4**- Let
**r**be the reminder when**b**is divided by 4, then units place of**a**will be equal to units place of^{b}**a**^{r}

- Let

**CYCLICITY TABLE FOR YOU**

Number | ^1 | ^2 | ^3 | ^4 | Cyclicity |
---|---|---|---|---|---|

2 | 2 | 4 | 8 | 6 | 4 |

3 | 3 | 9 | 7 | 1 | 4 |

4 | 4 | 6 | 4 | 6 | 2 |

5 | 5 | 5 | 5 | 5 | 1 |

6 | 6 | 6 | 6 | 6 | 1 |

7 | 7 | 9 | 3 | 1 | 4 |

8 | 8 | 4 | 2 | 6 | 4 |

9 | 9 | 1 | 9 | 1 | 2 |

So from the above table we get to know that the cyclicity of 7 is 4. So let is solve that question in which we had to find the unit digit of 2347

^{41}

And we have already discussed that we only need to find the unit digit of 7

^{41}and cyclicity of 7 is 4 so divide 41 we will get remainder 1 so the unit digit in 7

^{1}is 7 hence the unit digit of expression 2347

^{41}is 7.so we have learned finding the unit digit for any expression.