**High School Number Sense Lesson 54: Finding the Units Digit of a Number Raised to an Exponent**

Today's concept falls into the category of "easier than it looks." It is based upon the idea that any number raised to an exponent has the

**same units digit**as the original number's units digit raised to the same exponent. Instead of calculating the expression, the student needs to identify a pattern of the units digits of sequential powers of the number. This concept appeared

**10 times**last year, with a median placement at

**question # 47**.

**Number Dojo Level: TBD**

To visualize how to solve these problems, let's take a look at the first few exponents of each number. Pay attention to the units digits--and the pattern they create:

**0**

1 to any power is

**1**

2

^{1}=

**2**; 2

^{2}=

**4**; 2

^{3}=

**8**; 2

^{4}= 1

**6**; 2

^{5}= 3

**2**; ... (and the pattern repeats)

3

^{1}=

**3**; 3

^{2}=

**9**; 3

^{3}= 2

**7**; 3

^{4}= 8

**1**; 3

^{5}= 24

**3**; ... (and the pattern repeats)

4

^{1}=

**4**; 4

^{2}= 1

**6**; 4

^{3}= 6

**4**; ... (and the pattern repeats)

5 to any power ends in

**5**

6 to any power ends in

**6**

7

^{1}=

**7**; 7

^{2}= 4

**9**; 7

^{3}= 24

**3**; 7

^{4}= 240

**1**; 7

^{5}= 1680

**7**; ... (and the pattern repeats)

8

^{1}=

**8**; 8

^{2}= 6

**4**; 8

^{3}= 51

**2**; 8

^{4}= 409

**6**; 8

^{5}= 3276

**8**; ... (and the pattern repeats)

9

^{1}=

**9**; 9

^{2}= 8

**1**; 9

^{3}= 72

**9**; ... (and the pattern repeats)

**7**; 7 x 7 = 4

**9**; 9 x 7 = 6

**3**; 3 x 7 = 2

**1**; 1 x 7 =

**7**, etc.

**7, 9, 3, 1, 7...**It is perfectly acceptable to use your fingers to count out the exponent (term) you're on, while reciting in your head the elements of that pattern.

**Example 1:**

**7**

^{7}has a units digit of ___1. Figure out the pattern of the units digits of the powers of 7:

**7, 9, 3, 1, 7...**

2. The 7th number in that pattern is

**7**.

**Example 2:**

**The units digit of 9**

^{8}is ___1. Figure out the pattern of the units digits of the powers of 9:

**9, 1, 9, 1, ...**

2. The 8th number in that pattern is

**9**.

Example 3:

Example 3:

**15**

^{12}has a units digit of ___1. Ignore the tens digit and focus only on the

**5**.

2. This one is easy because 5 raised to any power has a units digit of

**5**.

**Example 4:**

**The units digit of 8**

^{11}is ___1. Figure out the pattern of the units digits of the powers of 8:

**8, 4, 2, 6, 8...**

2. The 11th number in that pattern is

**2**.

**Example 5:**

**14**

^{14}has a units digit of ___1. Ignore the tens digit and focus only on the

**4**.

2. Figure out the pattern of the units digits of the powers of 4:

**4, 6, 4, 6...**

3. The 14th number in that pattern will be a

**6**.

**Here's a free worksheet to help you practice UnitsDigit:**

unitsdigit.pdf |

**Up Next for High School: ExpandCoeff**