How to Multiply by Powers of 10

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  • 0:02 Multiplying by Powers of 10
  • 0:41 Using the Exponent in…
  • 1:26 Negative Exponents
  • 2:38 Multiplying Numbers
  • 4:36 Really Big & Really…
  • 5:35 Lesson Summary
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Lesson Transcript
Instructor: Thomas Higginbotham

Tom has taught math / science at secondary & post-secondary, and a K-12 school administrator. He has a B.S. in Biology and a PhD in Curriculum & Instruction.

Multiplying by powers of ten is one of the most commonly used math skills, and one of the easiest to learn. In this lesson, you will learn several tricks to help you remember this fundamental, versatile math skill.

Multiplying by Powers of 10

Multiplying by powers of 10 is a very common math skill. There are also just a few simple rules to do it!

The first thing you should know is what a power of 10 actually is. A power of 10 simply means the number of times 10 is multiplied by itself. For example, 10 x 10 is 10 to the second power, since we multiply 10 by itself 2 times. Following this pattern, 10 x 10 x 10 x 10 would be...? Hopefully, you realized this is 10 to the fourth power, since we multiply 10 by itself 4 times.

Using the Exponent in Powers of 10

Now, here is the great thing about powers of 10. The exponent (the number in superscript that indicates how many times you are multiplying 10 by itself) tells you exactly how many zeros follow the digit 1. For example:

102 is 100

The exponent is 2, and there are 2 zeros after the 1.

10 to the fourth power (104) is 10,000

The exponent is 4, and there are 4 zeros after the one. How easy is that?

Incidentally, 10 to the zero power also follows this pattern. 100 = 1. The exponent is 0, and there are no zeros after the 1. Numbers are awesome!

Negative Exponents

However, it all isn't so obviously nice and neat. 10 can also have a negative exponent; for example:

10 raised to the negative second power (10-2)

This means 1 divided by 102, or 1/100

10 to the negative fourth power (10-4) is 1 /104, or 1/10,000

There is a slight trick to it, but it's still pretty easy, huh?

Negative Exponents in Decimal Form

To represent 10 to a negative power as a decimal, we should go back to basics. 1/10 is one tenth or 0.1 when represented in decimal form. Notice there is one less zero to the left of the 1 (and to the right of the decimal point) than the exponent number. So:

10 -1 = 1/10 = one tenth = 0.1

Following the same pattern, 10 -2 = 1/102, or one one-hundredth (1/100) or 0.01

Try to write 10 to the negative fourth power! (Hint: It's 1/10,000 = 0.0001).

Multiplying Numbers

Everything we just learned isn't just for moving decimals around a 1! Let's look at some other numbers. Take 7 x 10, for instance. We all know that 7 x 10 = 70. We can see the 'number of zeroes' rule is the same when we multiply.

7 x 101 = 70, which is 7 with one zero after it. 7 x 100 = 700, we all know.

Same rule applies: 7 x 102 = 700, which is 7 with 2 zeroes after it!

When you are multiplying the powers of 10 and a number with digits after a decimal, we first focus on the decimal. We can move it to the left or right according to what powers of 10 we have. For example, 7.1805 x 10 = 71.805. This is the same as:

7.1805 x 101

We can solve this by simply moving the decimal point one place to the right. Similarly, 7.1805 x 100 is 718.05. Put in our exponent terms, it is:

7.1805 times 102

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