The Power Cycle in Maths

The power or indice of a number refers to the number of times a number is multiplied by itself. It is written as n^k, where n is the number and k is the number of times the number of times it should be multiplied by itself. It may seem that numbers appear random when they are raised to some power. However, on observation, each digit has a fixed pattern or cycle when they are raised to some powers. This cycle is termed the power cycle.

Types of Power Cycles

Before understanding the types of power cycles, some keywords should be kept in mind.

1. Frequency: Frequency or Cyclicity is the number of times after which the number starts repeating the cycle. Depending on the number, it can be one, two, or four.

2. Power Cycle: The power cycle is a set of all digits that appear for the number when finding its exponential before digits start repeating themselves. The number of elements of a power cycle set is called cyclicity or frequency.

Based on cyclicity, digits are organised into three categories.

1. Digits 0,1,5, and 6

2. Digits 4 and 9

3. Digits 2,3,7, and 8

Digits 0,1,5 and 6: For each of the digits, when the number is raised to any power, then it repeats itself. So, the cyclity for this category is 1.

Some examples to understand this category are as follows:

For 0, 0⁶ = 0, observe that the unit digit is 0.

For 5, 5² = 25, observe that the unit digit is 5.

For 1, 1⁴ = 1, observe that the unit digit is 1.

For 6, 6⁸ = 1679616, observe that the unit digit is 6.

Digits 4 and 9: For each of the digits, when the number is raised to any power, it repeats itself after one step. So, the cyclity for this category is 2.

Some examples to understand this category are as follows:

For 4, 4¹ = 4, observe that the unit digit is 4.

For 4, 4² = 16, observe that the unit digit is 6.

For 4, 4³ = 64, observe that the unit digit is 4.

Observe that 4 appeared in unit digit after a step.

For 9, 9² = 81, observe that the unit digit is 1.

For 9, 9³ = 729, observe that the unit digit is 9.

For 9, 9⁴ = 6561, observe that the unit digit is 1.

Observe that 1 appeared in unit digit after a step.

Digits 2,3,7, and 8: For each of the digits, when the number is raised to any power, it repeats itself after four steps. So, the cyclity for this category is 4.

Some examples to understand this category are as follows.

For 2, 2¹ = 2, observe that the unit digit is 2.

For 2, 2² = 4, observe that the unit digit is 4.

For 2, 2³ = 8, observe that the unit digit is 8.

For 2, 2⁴ = 16, observe that the unit digit is 6.

For 2, 2⁵ = 32, observe that the unit digit is 2.

Observe that 2 appeared in unit digit after three steps.

Similarly, for 3,

For 3, 3⁴ = 81, observe that the unit digit is 1.

For 3, 3⁵ = 243, observe that the unit digit is 3.

For 3, 3⁶ = 729, observe that the unit digit is 9.

For 3, 3⁷ = 2187, observe that the unit digit is 7.

For 3, 3⁸ = 6561, observe that the unit digit is 1.

Observe that 1 appeared in unit digit after three steps.

Similarly, for 7 and 8, the cyclicity is 7,9,3,1 and 8,4,2,4.

We can draw a characteristic table representing the cyclicity and power cycle for each number based on this information.

Conclusion

The power concept or the cyclicity of a number is a concept used in a number system to predict the unit digits of a number having an exponent, that is, ab. It can be used to compute unit digits of such numbers where exponents are quite large without finding the actual value. The power cycle does not depend on the number. Instead, it is dependent on the unit place digit of the number. The characteristic table helps to predict the unit place of the required exponent. Moreover, it can be noted that the digits having one or two frequencies will also repeat themselves with a frequency of four. So, one can say that the frequency of all the digits is four.