**Video Transcript**

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Transcript
**Exponents**

Once upon a time in a city in Persia, Maharaja Maini lived in a beautiful and ostentatious royal palace. One day, he realizes that his vault is overflowing with gold coins. He decides that he needs a larger vault to house his considerable treasure, and of course a greedy Maharaja like him wants to find the cheapest architect to work on his project. One offer in particular piques his interest. For payment, the architect requests 2 gold coins on the first day, 4 on the second, 8 on the third and so on. The payment continues to double each day until 30 days have passed. This seems to be a very good offer but the Maharaja's about to get a lesson in exponents. Let's take a look at how this payment system will unfold.

The architect gets paid daily. On the first day, the Maharaja pays the architect 2 gold coins. On the second day, he's gotta fork over double what he paid the previous day, for a total of 4 coins. By the 5th day, the Maharaja begins to wonder if his choice really was a good one.

The Maharaja decides he should calculate the payment for the rest of the month. He then starts calculating the payment for each day on a calendar. Hm...there isn't enough room in the box to write out what the Maharaja owes the architect on the 6th day.

Could there be a better way to write this? Let's see if we can find a pattern. You probably already know that multiplication is just repeated addition. For example, you already know a shorter way to write 2 plus 2 plus 2 plus 2 plus 2 plus 2. You can simply write 6 times 2. Both expressions equal 12. But, is there a shorter way to write repeated multiplication? There is! For example, we could write 2 times 2 times 2 times 2 times 2 times 2 as 2 to the sixth power which gives us 64.

Exponents offer a shorter way to write the repeated multiplcation of a single number. For example, let's take a look at 3 squared. In this example, 3 is our base and we call 2 the exponent, or power. The exponent represents the number of times a number is multiplied by itself. The expanded version of 3 to the 2nd power is 3 times 3 which equals 9, so 3 squared is 9. The typical way to express a number in general exponential form is 'a' to the nth power. This is the same as 'a' times 'a' times 'a' times 'a', 'n' times. Now the Maharaja can write what he owes the architect in the space provided on his calendar!

Let's see if we can rewrite what the Maharaja owes the architect for the first 5 days. On day 1, the Maharaja owes the architect 2 gold coins which we can write this as 2 to the first power. On day 2, the Maharaja owes the architect 2 squared gold coins. On day 3, the Maharaja owes the architect 2 cubed gold coins. Are you sensing a pattern? How could we express how many gold coins the Maharaja owes the architect on day 5? That's right! We can write it as 2 to the 5th power!

By the 30th day, Maharaja Maini, using 2 as his base and 30 as his exponent calculates what he owes the architect on day 30. After paying the architect, Maharaja Maini senses there must be a mistake his vault is expanded, just like he wanted. But now the Maharaja's got a new problem.