Exponents and Simplifying Expressions

Motivation for Exponents

Which would you choose?

You are given an option of being paid either

• $1,000,000 or

• for thirty days receiving one cent the first day and each succeeding day double the amount of the previous day.

How much would you receive on the 30^th day?

The double of an amount is the same as multiplying an amount by two. So, the daily amounts in cents for each day would be:

Day 1               1¢
Day 2               2¢
Day 3               2 × 2 = 4¢
Day 4               2 × 2 × 2 = 8¢
Day 5               2 × 2 × 2 × 2 = 16¢
Day 6               2 × 2 × 2 × 2 × 2 = 32¢
Day 7               2 × 2 × 2 × 2 × 2 × 2 = 64¢

This is getting to be a lot of multiplications to write. We use exponents to write the problem for each day in a shorter form. Since Day 7 has two used as a factor six times, we may write the problem as

Day 7               2⁶ = 2 × 2 × 2 × 2 × 2 × 2 = 64¢

Then Day 8 would use another factor of 2 for

Day 8               2⁷ = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128¢
Day 9               2⁸ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256¢
Day 10           2⁹ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 512¢

We note that the number of two's used as a factor is one less than the number of the day, so

Day 30           2²⁹.

This says that we would need to multiply 29 two's to find the amount of money in cents for the thirtieth day. Determine this amount then decide which choice would be the best choice.

Solution

The above problem motivates why exponents are used as a shortcut to setup problems. We now give more examples involving exponents.

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