How to Multiply Quickly with Exponents
You can multiply very quickly when you understand the concept of exponents. Here's an old question whose answer may surprise you: Suppose you took a job that paid you just 1 penny the first day, 2 pennies the second day, 4 pennies the third day, and so on, doubling the amount every day, like this:
1 2 4 8 16 32 64 128 256 512 ...
As you can see, in the first ten days of work, you would've earned a little more than $10 (actually, $10.23 — but who's counting?). How much would you earn in 30 days? Your answer may well be, "I wouldn't take a lousy job like that in the first place." At first glance, this looks like a good answer, but here's a glimpse at your second ten days' earnings:
... 1,024 2,048 4,096 8,192 16,384 32,768 65,536 131,072 262,144 524,288 ...
By the end of the second 10 days, your total earnings would be over $10,000. And by the end of 30 days, your earnings would top out around $10,000,000! How does this happen? Through the magic of exponents (also called powers). Each new number in the sequence is obtained by multiplying the previous number by 2:
As you can see, the notation 24 means multiply 2 by itself 4 times.
You can use exponents on numbers other than 2. Here's another sequence you may be familiar with:
1 10 100 1,000 10,000 100,000 1,000,000...
In this sequence, every number is 10 times greater than the number before it. You can also generate these numbers using exponents:
This sequence is important for defining place value, the basis of the decimal number system. It also shows up with decimals and scientific notation.