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.

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.

Number | Cyclicity | Power Cycle |

1 | 1 | 1 |

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

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

4 | 2 | 4,6 |

5 | 1 | 5 |

6 | 1 | 6 |

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

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

9 | 2 | 9,1 |

10 | 1 | 0 |

**Number 0 :**

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

Power of 0 | Value |

0^1 | 0 |

0^2 | 0 |

0^3 | 0 |

0^4 | 0 |

0^5 | 0 |

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 1 | Value |

1^1 | 1 |

1^2 | 1 |

1^3 | 1 |

1^4 | 1 |

1^5 | 1 |

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 2 | Value |

2^1 | 2 |

2^2 | 4 |

2^3 | 8 |

2^4 | 16 |

2^5 | 32 |

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 3 | Value |

3^1 | 3 |

3^2 | 9 |

3^3 | 27 |

3^4 | 81 |

3^5 | 243 |

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 4 | Value |

4^1 | 4 |

4^2 | 16 |

4^3 | 64 |

4^4 | 256 |

4^5 | 1024 |

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 1 | Value |

5^1 | 5 |

5^2 | 25 |

5^3 | 125 |

5^4 | 625 |

5^5 | 3125 |

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 6 | Value |

6^1 | 6 |

6^2 | 36 |

6^3 | 216 |

6^4 | 1296 |

6^5 | 7776 |

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 7 | Value |

7^1 | 7 |

7^2 | 49 |

7^3 | 343 |

7^4 | 2401 |

7^5 | 16807 |

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 8 | Value |

8^1 | 8 |

8^2 | 64 |

8^3 | 512 |

8^4 | 4096 |

8^5 | 32768 |

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 9 | Value |

9^1 | 9 |

9^2 | 81 |

9^3 | 729 |

9^4 | 6651 |

9^5 | 59859 |

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)**

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 Concept | Key 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. |

**Number 0 :**

**Answer:**

Look at the unit place of number 24**0**. Unit’s place digit is **0**.

**Power Cycle of 0 : (0)**

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

**Answer:**

Look at the unit place of number 34**0**. 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 :**

**Answer:**

Look at the unit place of number 12**1**. Unit’s place digit is 1.

**Power Cycle of 1 : (1)**

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

**Answer:**

Look at the unit place of number 79**1**. Unit’s place digit is 1.

**Power Cycle of 1 : (1)**

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

**Number 2 :**

**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**

Remainder | Unit Place digit |

1 | 2 |

2 | 4 |

3 | 8 |

0 | 6 |

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.

Look at the unit place of number 122**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: 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**

Remainder | Unit Place digit |

1 | 2 |

2 | 4 |

3 | 8 |

0 | 6 |

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

**Number 3 :**

**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.

Remainder | Unit Place digit |

1 | 3 |

2 | 9 |

3 | 7 |

0 | 1 |

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