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Saxon Math 8/7 Homeschool: Online Textbook Help34 chapters | 191 lessons

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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 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.

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!

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?

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).

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

Now, 7.1805 x 1000 is...? You got it! 7180.5, or the factor with a decimal point moved three places to the right.

With negative powers of 10, the rule is the same: move the decimal point the number of places indicated by the exponent.

7 times 10 -1 = 0.7. 7 x 10-4 = 0.0007

Multiplying by negative exponents, the decimal point moves to the left; multiplying by positive exponents, the decimal point moves to the right.

We started with reasonably sized numbers so that the reasons behind the rules would not only be easy to understand, but also easy to remember. If you forget the rules, you can go back to one of our examples and uncover the rules for yourself.

Now that you're ready, we will take a look at two examples: one really big and one really small.

Avogadro's number is 6.023 x 1023. What would this be in decimal form? Simple, take 6.023 and move the decimal to the right 23 places, and where the non-zero digits are, add in 0s. Your final answer would be this: 602,300,000,000,000,000,000,000!

A dust particle has a mass of approximately 7.53 x 10-13 grams. What is this in decimal form? Move the decimal point 13 places to the LEFT, which gives this: 0.000000000000753 grams.

Multiplying by a **power of 10** is easy using just a few basic rules:

1. The number of zeroes for a power of 10 is equal to the power to which 10 is raised:

105 = 100,000

2. When multiplying a factor by a power of 10, move the decimal point the same number of places as indicated by the exponent:

4.1234 x 103 = 4123.4

This works for negative exponents as well:

4.1234 x 10-3 = 0.0041234

For negative exponents, move the decimal point to the left. For positive exponents, move the decimal point to the right.

3. If you forget the rules, go back to basics with a simple problem and work backwards to find your pattern.

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