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Ones digit of an exponent

Ones digit of an exponent, through following a pattern

This math skill involves identifying the ones digit of a large exponent by recognizing and applying a repeating pattern of ones digits in successive powers of the base number. The pattern, once identified, simplifies finding the ones digit without needing to compute the full exponentiation.

Learn ones digit of an exponent

Teaching Transcript

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If we calculated 63 to the power of 81, what would its ones digit be?

Obviously it would be an incredibly large number, so we won't try to calculate it. Any exponent is just the base number multiplied that many times. We can see that the ones digit of each multiplication is 3.

If we multiply any number ending in 3 by any number ending in 3, we get a number ending in 9. Likewise, any number ending in 9, multiplied by 3 will end in a 7. Ones digits of 3 and 7 will have a ones digit of 1, and 1 by 3 will have a ones digit of 3.

Notice that we are now back to the start of the pattern, with a ones digit of 3.

If we express this as exponents, 3 to the power of 1 has a ones digit of 3, then for each subsequent power, the ones digit is 9, 7, 1, then 3 again.

The pattern repeats every fourth power, so 3 to the power of 80 will have the same ones digit as any multiple of 4, which is 1.

And if we continue one more, that's a ones digit of 3

We've been looking at a base of 3, but it's only the ones digit of the base that matters, so a base of 63 will follow the same pattern. 63 to the power of 81 would also have a ones digit of 3.

Skill Practice

Ones digit of an exponent, through following a pattern

Test your understanding of ones digit of an exponent by practicing it! Work through the below exercises to use it in practice.