What is power cycle in maths?

Power Cycle (Xn - Unit Place Digit) is a way to find out the repeating pattern of numbers that appear in the unit's place (the rightmost digit) when we raise a particular number to different powers.

In order to ensure that you are fully prepared for any exam, it's important to develop a deep understanding of power cycles for all numbers 0 to 9. By doing so, you will be able to easily solve any problem that is presented to you during the exam, and even be able to articulate your thought process orally. With this knowledge, you will not only be able to perform well on exams, but will also have a solid foundation for future problem-solving endeavors in the field of maths.

Power Cycles for all Numbers (Focus on Unit place digit only)

If you can remember power cycle values that would be great however even if you are unable to remember it, we will look at technique to calculate power cycle of number in less than 1 minute during exam as well and you can solve problem.

Powercycle of numbers 0-10

NumberCyclicityPower Cycle
111
242,4,8,6
343,9,7,1
424,6
515
616
747,9,3,1
848,4,2,6
929,1
1010

Finding the Power Cycle of numbers

Number 0 :

Let us calculate values for 0^1 to 0^5

Power of 0Value
0^10
0^20
0^30
0^40
0^50

So as you can observe,

Value of 0^1 to 0^5 is 0 only. So Unit place digit is 0 for any power of 0

Power Cycle for 0 : (0)

Number 1 :

Let us calculate values for 11 to 15

Power of 1Value
1^11
1^21
1^31
1^41
1^51

So as you can observe:

Value of 11 to 15 is 1 only. So Unit place digit is 1 for any power of 1

Power Cycle for 1 : (1)

Number 2 :

Let us calculate values for 2^1 to 2^5

Power of 2Value
2^12
2^24
2^38
2^416
2^532

So as you can observe:

Unit place digit of 2^1 to 2^5 is in order 2, 4, 8, 6 and it will keep repeating as 2, 4, 8, 6

Power Cycle for 2 : (2, 4, 8, 6)

Number 3 :

Let us calculate values for 3^1 to 3^5

Power of 3Value
3^13
3^29
3^327
3^481
3^5243

So as you can observe:

Unit place digit of 3^1 to 3^5 is in order 3, 9, 7, 1 and it will keep repeating as 3, 9, 7, 1

Power Cycle for 3 : (3, 9, 7, 1)

Number 4 :

Let us calculate values for 41 to 45

Power of 4Value
4^14
4^216
4^364
4^4256
4^51024

So as you can observe:

Unit place digit of 4^1 to 4^5 is in order 4, 6 and it will keep repeating as 4, 6

Power Cycle for 4 : (4, 6)

Number 5:

Let us calculate values for 5^1 to 5^5

Power of 1Value
5^15
5^225
5^3125
5^4625
5^53125

So as you can observe:

Unit place digit of 5^1 to 5^5 is 5 only. So Unit place digit is 5 for any power of 5

Power Cycle for 5 : (5)

Number 6:

Let us calculate values for 6^1 to 6^5

Power of 6Value
6^16
6^236
6^3216
6^41296
6^57776

So as you can observe:

Unit place digit of 6^1 to 6^5 is 6 only. So Unit place digit is 6 for any power of 6

Power Cycle for 6 : (6)

Number 7 :

Let us calculate values for 7^1 to 7^5

Power of 7Value
7^17
7^249
7^3343
7^42401
7^516807

So as you can observe:

Unit place digit of 7^1 to 7^5 is in order 7, 9, 3, 1 and it will keep repeating as 7, 9, 3, 1

Power Cycle for 7 : (7, 9, 3, 1)

Number 8 :

Let us calculate values for 81 to 85

Power of 8Value
8^18
8^264
8^3512
8^44096
8^532768

So as you can observe:

Unit place digit of 8^1 to 8^5 is in order 8, 4, 2, 6 and it will keep repeating as 8, 4, 2, 6

Power Cycle for 8 : (8, 4, 2, 6)

Number 9 :

Let us calculate values for 9^1 to 9^5

Power of 9Value
9^19
9^281
9^3729
9^46651
9^559859

So as you can observe:

Unit place digit of 9^1 to 9^5 is in order 9, 1 and it will keep repeating as 9, 1

Power Cycle for 9 : (9, 1)

Calculating the Power of a Number

Once you have found the power cycle for a number, you can use it to calculate the power of that number. To do this, simply look for the desired power in the power cycle. For example, to calculate the power of 5 to the power of 10, you would look for the 10th number in the power cycle for 5.

Power Cycle ConceptKey Points
Basics of Exponents- Exponents represent the number of times a base number is multiplied by itself. - The base number is the number being multiplied repeatedly. - The exponent indicates the number of times the base number is multiplied by itself.
Multiplying with Exponents- When multiplying numbers with the same base, add their exponents. - This simplifies calculations and allows for easier manipulation of larger numbers.
Exploring Higher Powers- Raising a number to the power of zero equals 1. - Negative exponents indicate the reciprocal of the number.
The Power Cycle in Equations- Utilize the power cycle to simplify and solve equations. - Transform equations using the properties of exponents for more manageable problem-solving.

Problems based on Power cycle

Number 0 :

Q1: What will be unit's place digit for 240^143 ?

Answer:

Look at the unit place of number 240. Unit’s place digit is 0.

Power Cycle of 0 : (0)

Answer is unit's place digit for 240^143 will be 0.

Q2: What will be unit's place digit for 340^4103 ?

Answer:

Look at the unit place of number 340. Unit’s place digit is 0.

Power Cycle of 0 : (0)

Answer is unit's place digit for 340^4103 will be 0.

Number 1 :

Q3: What will be unit's place digit for 121^53 ?

Answer:

Look at the unit place of number 121. Unit’s place digit is 1.

Power Cycle of 1 : (1)

Answer is unit's place digit for 121^53 will be 1.

Q4: What will be unit's place digit for 791^5643 ?

Answer:

Look at the unit place of number 791. Unit’s place digit is 1.

Power Cycle of 1 : (1)

Answer is unit's place digit for 7915643 will be 1

Number 2 :

Q5: What will be unit's place digit for 2^33 ?

Answer:

Look at the unit place of number 2. Unit’s place digit is 2.

Power Cycle of 2 : (2, 4, 8, 6)

There are total 4 values which keep repeating always for power of 2. Now look at index which is to be identified: 33

As 4 numbers keep on repeating for power cycle of 2, we need to divide 33 by 4 and identify remainder of it so that we can understand what can be unit place number.

Index to be found / Size of power cycle 33,

Quotient = 8 and Remainder = 1

RemainderUnit Place digit
12
24
38
06

You don’t need to remember this table you just need to make sure as you know pattern of power cycle you have to reach till index number.

Like in this case:

To reach 33 and you have size of 4

4, 8, 12…..32 so 32nd index would be last number in power cycle that is 6

33rd index would have 1st number in power cycle that is 2

Answer is unit's place digit for 2^33 will be 2.

Q6: What will be unit's place digit for 1222^438

Look at the unit place of number 1222. Unit’s place digit is 2.

Power Cycle of 2 : (2, 4, 8, 6)

There are total 4 values which keep repeating always for power of 2. Now look at index which is to be identified: 438

As 4 numbers keep on repeating for power cycle of 2, we need to divide 438 by 4 and identify remainder of it so that we can understand what can be unit place number.

Index to be found / Size of power cycle 438 /4,

Quotient = 109 and Remainder = 2

RemainderUnit Place digit
12
24
38
06

Answer is unit's place digit for 1222^438 will be 4

Number 3 :

Q7: What will be unit's place digit for 3^36 ?

Answer:

Look at the unit place of number 3. Unit’s place digit is 3.

Power Cycle of 3 : (3, 9, 7, 1)

There are total 4 values which keep repeating always for power of 3. Now look at index which is to be identified: 36

As 4 numbers keep on repeating for power cycle of 3, we need to divide 36 by 4 and identify remainder of it so that we can understand what can be unit place number.

Index to be found / Size of power cycle = 36/4

Quotient = 9 and Remainder = 0

Whenever remainder is 0 it is last digit in power cycle.

RemainderUnit Place digit
13
29
37
01