How do you write a power of 10?

Video transcript

- [Instructor] In this video, I'm going to introduce you to a new type of mathematical notation that will seem fancy at first, but hopefully you'll appreciate is pretty useful and also pretty straightforward. So let's just start with some things that we already know. So I could have just a 10. I could take two 10s and multiply them together, so 10 times 10, which you know is equal to 100. I could take three 10s and multiply them together. 10 times 10 times 10 is equal to 1000. And I could do this with any number of 10s. But at some point, if I'm doing this with enough 10s, it gets pretty hard to write. So for example, let's say I were to do this with 10 10s. So if I were to go 10 times 10 times 10 times 10, that's four, that's five, that's six, that's seven, that's eight, that's nine, that is 10 10s. Let's see, one, two, three, four, five, six, seven, eight, nine, 10. This is going to be equal to, even the number that it's equal to is going to be quite hard to write. It's going to be one followed by 10 zeroes. One, two, three, four, five, six, seven, eight, nine, 10. We put the commas there so it's just a little bit easier to read. This right over here is 10 billion, and it's already getting kinda hard to write, and imagine if we have 30 10s that we were multiplying together. So mathematicians have come up with a notation and some ideas to be able to write things like this a little bit more elegantly. So the way they do this is through something known as exponents. Exponents. And so 10 times 10, we can rewrite as being equal to, if I have two 10s and I'm multiplying them together, I could write this as 10 to the second power. That's how someone would say it. They would say 10 to the second power. That looks fancy, but all that means is let's take two 10s and multiply them together and we're going to get 100. Just so you're familiar with some of the parts of this, the two would be called the exponent and the 10 would be the base. So 10 to the second power is 10 times 10 is equal to 100. So how would you write 10 times 10 times 10 or 1000? How would you write that using exponents? Pause this video and see if you can figure that out. Well, as exactly as you might have imagined, we're taking a certain number of 10s and we see we're taking three 10s and we're multiplying them together. So this would be 10 to the third power. 10 is the base, three is the exponent. We would read this as 10 to the third power. If you ever saw 10 to the third power, that means hey, let me multiply 10 times 10 times 10. That's the same thing as 1000. So this is really another way of writing 1000. And what about this number here, 10 billion? What's a way that we could write it using exponents? Pause the video and figure that out. Well as you might have imagined, we were taking 10 10s and multiplying them together. This is 10 to the 10th power. And we can go the other way around. If someone were to walk up to you on the street and say what is 10 to the fifth power? What is that? What number that you're probably familiar with would this be? Well this would mean that we're going to take five 10s and multiply them together. So 10 times 10 times 10 times 10 times 10. And so 10 times 10 is 100, 100 times 10 is 1000, 1000 times 10 is 10000, 10000 times 10 is equal to 100000. So there you have it. You have the basics of exponents when we're dealing with 10, and I know what you were thinking. Can I put another number here instead of 10? And the simple answer is, you can, but we're not gonna cover that just yet in this video.

The exponent (or index or power) of a number says
how many times to use the number in a multiplication.

102 means 10 × 10 = 100

(It says 10 is used 2 times in the multiplication)

Example: 103 = 10 × 10 × 10 = 1,000

• In words: 103 could be called "10 to the third power", "10 to the power 3" or simply "10 cubed"

Example: 104 = 10 × 10 × 10 × 10 = 10,000

• In words: 104 could be called "10 to the fourth power", "10 to the power 4" or "10 to the 4"

You can multiply any number by itself as many times as you want using this notation (see Exponents), but powers of 10 have a special use ...

Powers of 10

"Powers of 10" is a very useful way of writing down large or small numbers.

Instead of having lots of zeros, you show how many powers of 10 will make that many zeros

Example: 5,000 = 5 × 1,000 = 5 × 103

5 thousand is 5 times a thousand. And a thousand is 103. So 5 times 103 = 5,000

Can you see that 103 is a handy way of making 3 zeros?

Scientists and Engineers (who often use very big or very small numbers) like to write numbers this way.

Example: The Mass of the Sun

The Sun has a Mass of 1.988 × 1030 kg.

It is too hard to write 1,988,000,000,000,000,000,000,000,000,000 kg

(And very easy to make a mistake counting the zeros!)

Example: A Light Year (the distance light travels in one year)

It is easier to use 9.461 × 1015 meters, rather than 9,461,000,000,000,000 meters

It is commonly called Scientific Notation, or Standard Form.

Other Way of Writing It

Sometimes people use the ^ symbol (above the 6 on your keyboard), as it is easy to type.

Example: 3 × 10^4 is the same as 3 × 104

• 3 × 10^4 = 3 × 10 × 10 × 10 × 10 = 30,000

Calculators often use "E" or "e" like this:

Example: 6E+5 is the same as 6 × 105

• 6E+5 = 6 × 10 × 10 × 10 × 10 × 10 = 600,000

Example: 3.12E4 is the same as 3.12 × 104

• 3.12E4 = 3.12 × 10 × 10 × 10 × 10 = 31,200

The Trick

While at first it may look hard, there is an easy "trick":

The index of 10 says ...

... how many places to move the decimal point to the right.

Example: What is 1.35 × 104 ?

You can calculate it as: 1.35 x (10 × 10 × 10 × 10) = 1.35 x 10,000 = 13,500

But it is easier to think "move the decimal point 4 places to the right" like this:

Negative Powers of 10

Negative? What could be the opposite of multiplying? Dividing!

A negative power means how many times to divide by the number.

Example: 5 × 10-3 = 5 ÷ 10 ÷ 10 ÷ 10 = 0.005

Just remember for negative powers of 10:

For negative powers of 10, move the decimal point to the left.

So Negatives just go the other way.

Example: What is 7.1 × 10-3 ?

Well, it is really 7.1 x (1/10 × 1/10 × 1/10) = 7.1 × 0.001 = 0.0071

But it is easier to think "move the decimal point 3 places to the left" like this:

Try It Yourself

Enter a number and see it in Scientific Notation:

Now try to use Scientific Notation yourself:

Summary

The index of 10 says how many places to move the decimal point. Positive means move it to the right, negative means to the left. Example:

How do you write 10 to the first power?

What is 1 as a power of 10?