To solve this, we’ll begin examining smaller powers and look for a pattern.

(the units digit is 7)

(the units digit is 9)

(the units digit is 3)

Aside: Since these powers increase quickly, it’s useful to notice that we need only multiply the units digit each time. For example, the units digit of is the same as the units digit of . Similarly, the units digit of is the same as the units digit of .

So, once we know that the units of is 9, we can find the units digit of by multiplying 9 by 7 to get 63. So the units digit of is 3.

To find the units digit of , we’ll multiply 3 by 7 to get 21. So the units digit of is 1.

When we start listing the various powers, we can see a pattern emerge:

The units digit of is 7

The units digit of is 9

The units digit of is 3

The units digit of is 1

The units digit of is 7

At this point, we should recognize that the pattern begins to repeat. The pattern goes: 7-9-3-1-7-9-3-1-7-9-3-1-…

Since the pattern repeats itself every 4 powers, we say that the “cycle” equals 4

Now comes an important observation:

The units digit of is 7

The units digit of is 9

The units digit of is 3

The units digit of is 1

The units digit of is 7

The units digit of is 9

The units digit of is 3

The units digit of is 1

The units digit of is 7

The units digit of is 9

The units digit of is 3

The units digit of is 1. . .etc.

As you can see, since the cycle = 4, the units digit of will be 1whenever k is a multiple of4.

Now to find the units digit of , all we need to do is recognize that the units digit of is 1 (since 44 is a multiple of 4).

From here, we’ll just continue with our pattern:

The units digit of is 1

The units digit of is 7

The units digit of is 9

The units digit of is 3 . . . etc.

So, the units digit of is 7, which means the answer is D.

If you’d like to practice, you can answer these two questions:

What is the units digit of ?

What is the units digit of ?

(The answers can be found at the very bottom of this post)