### Theory:

In the last topic, we have learned how to write a number in expanded form and exponential form.

Now, we shall learn how to find the unit digit of a number which is in exponential form.

Let us consider $$6^3 = 6 \times 6 \times 6$$ $$= 216$$. Here, the unit digit is $$6$$.

Similarly, let us consider $$5^5 = 5 \times 5 \times 5 \times 5 \times 5$$ $$= 3125$$. Here, the unit digit is $$5$$.

It is easier to determine the unit digit of a number whose exponential is small.

Is it possible to find the unit digit of $$26^{34}$$, $$45^{20}$$, $$20^{33}$$?

It isn't easy to find the unit digit by expanding the exponential form. But, there are specific patterns to find the unit digit.

We know that the numbers ends with either of the following digits:

$$1$$ or $$2$$ or $$3$$ or $$4$$ or $$5$$ or $$6$$ or $$7$$ or $$8$$ or $$9$$ or $$0$$.

There are some groups of numbers which following certain patterns when we find the exponential value of the numbers. Now we will learn those groups in detail by observing its pattern.
Let us observe the pattern for the numbers $$0$$, $$1$$, $$5$$ and $$6$$.
Pattern for the number $$0$$:

$$0^2 = 0 \times 0 = \underline{0}$$

$$0^3 = 0 \times 0 \times 0 = \underline{0}$$

$$0^4 = 0 \times 0 \times 0 \times 0 = \underline{0}$$

Pattern for the number $$1$$:

$$1^2 = 1 \times 1 = \underline{1}$$

$$1^3 = 1 \times 1 \times 1 = \underline{1}$$

$$1^4 = 1 \times 1 \times 1 \times 1 = \underline{1}$$

Pattern for the number $$5$$:

$$5^2 = 5 \times 5 = 2\underline{5}$$

$$5^3 = 5 \times 5 \times 5 = 12\underline{5}$$

$$5^4 = 5 \times 5 \times 5 \times 5 = 62\underline{5}$$

Pattern for the number $$6$$:

$$6^2 = 6 \times 6$$ $$= 3\underline{6}$$

$$6^3 = 6 \times 6 \times 6$$ $$= 21\underline{6}$$

$$6^4 = 6 \times 6 \times 6 \times 6$$ $$= 129\underline{6}$$

Here, in the above patterns, we can see that, any number in the exponential form of $$0^x$$, $$1^x$$, $$5^x$$ and $$6^x$$, the unit digit will be the same number where $$x$$ is positive number.
Example:
$$66^2 =$$ $$(11 \times 6)^2 =$$ $$11^2 \times 6^2$$ $$= 121 \times 36$$ $$= 435\underline{6}$$

Here, the unit digit is $$6$$.
Let us observe the pattern for the numbers $$4$$ and $$9$$.
Pattern for the number $$4$$:

$$4^1 = \underline{4}$$   (Odd pattern)

$$4^2 = 4 \times 4$$ $$= 1\underline{6}$$   (Even pattern)

$$4^3 = 4 \times 4 \times 4$$ $$= 6\underline{4}$$    (Odd pattern)

$$4^4 = 4 \times 4 \times 4 \times 4$$ $$= 25\underline{6}$$   (Even pattern)

$$4^5 = 4 \times 4 \times 4 \times 4 \times 4$$ $$= 102\underline{4}$$   (Odd pattern)

Here, we can see that if the power is odd, then the unit digit of the resultant is $$4$$. If  the power is even, then the unit digit of the resultant is $$6$$.

Pattern for the number $$9$$:

$$9^1 = \underline{9}$$   (Odd pattern)

$$9^2 = 9 \times 9$$ $$= 8\underline{1}$$   (Even pattern)

$$9^3 = 9 \times 9 \times 9$$ $$= 72\underline{9}$$    (Odd pattern)

$$9^4 = 9 \times 9 \times 9 \times 9$$ $$= 656\underline{1}$$   (Even pattern)

$$9^5 = 9 \times 9 \times 9 \times 9 \times 9$$ $$= 5904\underline{9}$$   (Odd pattern)

Here, we can see that if the power is odd, then the unit digit is $$9$$. If  the power is even, then the unit digit is $$1$$.
Example:
Find the unit digit of the number $$19^{16}$$.

Solution:

Here, the unit digit of base $$19$$ is $$9$$ and power is $$16$$ (even power).

For base ending with $$9$$ having an even power, the unit digit is $$1$$.

Therefore, the unit digit is $$1$$.