Multiplying With Exponents

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  • 0:01 Exponents
  • 1:00 Using Exponents to Multiply
  • 4:10 Example 1
  • 4:42 Example 2
  • 5:45 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn how you can use exponents to help you multiply numbers together. Learn what kinds of numbers you can multiply together with exponents and how to write them.


Exponents are used a lot in math and in the sciences. Exponents are the powers that numbers are raised to. You can tell if a number is an exponent if it is written in the upper right corner of a number. When you can't write a number as a smaller number written in the upper right corner, like when you are typing things out on a computer, then you write the exponent using the little caret symbol as in 3^4. In this example, the 4 is the exponent, and the 3 is the number that is being raised to the power of 4.

Who uses exponents? Mathematicians, of course, and scientists. Scientists use exponents a lot when describing numbers that are really large or very small. You might see the scientific number 3 x 10^8. This tells you that you have a 3 followed by 8 zeroes. Now that's a really large number!

Using Exponents to Multiply

Now, how did we know that the 10^8 has 8 zeroes? Because exponents are actually another way to write out numbers that are being multiplied together. So 10^8 actually means 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10. The exponent tells you to multiply the number that many times. So an exponent of 8 means you multiply the number eight times. Since our number is 10, we multiply the 10 eight times. What do we get when we do that? We get a 1 followed by eight zeroes: 100,000,000.

Let me ask you, what do you think the 3^4 is telling you? Is it telling you to multiply the 4 three times, or is it telling you to multiply the 3 four times? Is this a trick question? Maybe. Well, which one is your exponent? The 4. That's right. So which one is your number then? 3. If 3 is your number and 4 is your exponent, then 3^4 is telling you to multiply the 3 four times: 3^4 = 3 * 3 * 3 * 3.

We can easily go the other way too. If we see a problem, such as 2 * 2 * 2, we can turn this into an exponent problem. We write our number 2 down. Then we count how many times it is being multiplied together. This gives us our exponent. The 2 is being multiplied three times, so our exponent is 3. We can write 2 * 2 * 2 = 2^3.

Do you notice a pattern here as far as our numbers are concerned? Our numbers have to be the same for us to write them using exponents. If we have different numbers being multiplied together, we can't use exponents to show the multiplication. But, if we have several different numbers that are each multiplied by themselves several times, then we can still use exponents to rewrite each set of numbers.

For example, we can rewrite the problem 2 * 2 * 2 * 4 * 4 * 4 * 4 as 2^3 and 4^4. We have two different numbers with each number being multiplied by itself several times. The 2 is being multiplied by itself 3 times. We can rewrite that using exponents as 2^3. The 4 is being multiplied by itself 4 times, so we can rewrite that as 4^4.

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