Here's part 2 for our math quill guide for unit 5. I mentioned learning about entering numbers is scientific notation. Let's see how we can do this. To enter a number in scientific notation. You'll want to type in the leading number or decimal followed by the asterisk symbol. This asterisk symbol will create a dot or a multiplication in between our first number and our power of 10. Be sure you don't type x for the multiplication sign. We'll think it's a variable instead. Next, you'll type in the number 10 followed by the carrot symbol and then the exponent of 5. So now you try it out. You can follow these keystrokes to enter this number in scientific notation and then, try these on your own. Enter your answers here and good luck.
For the second one, you want to type in 3.06. Then, use the asterisk symbol, the number 10, the carrot symbol, and then 34 to get the exponent. For this last number in scientific notation, we start by entering the decimal, followed by asterisk symbol, and the number 10. Then, we type the carrot key and negative sign and 9 to give us our final exponent. I hope you're starting to feel like a Math Quill pro if you're getting all these correct.
We use scientific notation to write extremely large numbers or small numbers, let's review that real quick. At the beach, we could think about measuring a tiny grain of sand, that grain would measure this many meters. We can rewrite this number in scientific notation, using what we learned from Unit 1. For scientific notation, the absolute value of our first number, a, must be greater than or equal to 1, and less than 10. So we know we need to move the decimal to the 6. We'll have 6 and 25 hundredths. Next, we need a power of 10 so we can move this decimal back to its original position. I know this decimal should move five places to the left in order to get this number. So my exponent is negative should make a ton of sense now. Multiplying by a negative power of ten makes a number smaller because we're really dividing the number by a power of 10. And, if we carried out the math and moved the 10 to the denominator and then divided by 100,000 we would get this decimal. And, remember a positive exponent would make sense here. Positive powers of ten make numbers larger but this one's smaller.
Let's make sure you've mastered this skill. Write these different numbers in scientific notation. The first is the distance from the earth to the moon. The second is the distance light travels in a year. And this last small number is the wavelength of red light. A wavelength of light is a measurement from the crest to the crest, or a trough to a trough. If you're interested about light, try looking some things up. Write your answers in scientific notation in these boxes, and don't include the unit. I've got that written for you. Also, be sure you don't round any of the numbers. You want the most accurate number with all the decimals after the decimal point. Good luck.
You've mastered scientific notation for getting these right. If you had trouble with one or two of them, that's okay too. For this number, the a value would need to be 2.389, and I would have to move this decimal five places to the right, to get back to my original number. This number is also really large, so we know we're going to need a positive power. If I think about moving this decimal 12 places to the right, I end up with my original number. The last one, the decimal point needs to come after the 7. So if it were after the 7, I would need a negative exponent to move this decimal back to the left 7 spaces, and I'd get this number. And we now multiplying by a negative power 10 really divides this by a power of 10, so our number becomes smaller.
What's great about knowing scientific notation an the property of exponents, is that it can make these types of calculations a lot easier. We can actually perform this without using a calculator. First I can change each of the numbers into scientific notation and I need to make sure that I'm multiplying these together, we still have multiplication between each of the numbers. This numerator and this denominator are really just products of factors, so I can regroup things. I'm going to regroup my powers of 10 next to one another, and put the numbers next to one another. And here's where our exponent properties come into play. I want you to simplify these powers of 10. What would be the single exponent on the power of 10, if we simplified this part of the fraction? Write your answer in this box.
If you said 9, excellent work. I've copied this part of the fraction over here. Working with bases of 10, I can use my exponent rules to figure out what the power should be. Here are multiplying bases, so we add the exponents. 5 plus negative 3 equals 2, and negative 4 plus negative 3 equals negative 7. Next I can subtract my exponent since I'm doing division of the bases. So 10 to the 2 minus negative 7, or 10 to the 9th. This also makes sense because we know negative exponents wind up in the opposite position. So 10 to the negative 7th would wind up being 10 to the positive 7th in the numerator. 10 to the 2nd and
Our last step is to simplify common factors that appear in the numerator and the denominator. I can divide each of these numbers by 1 and 2 10ths, and I'll have divide 1 and 8 10ths and I get 9 10ths. Our number isn't quite yet in scientific notation, we'll need to move this decimal one place to right. So what should this last exponent be? Take some time to think about, and think about writing this number out if it helps you.
The exponent should be a positive 8. Great thinking if you got it right. If you multiply 9 10ths times 10 to the 9th, we would get 900,000,000. And 9 followed by eight 0s. In order to get this number back in scientific notation, our decimal point would have to be right after the 9. This means the power on the 10 must be an 8. Our decimal point is here, and if we move it 8 places to the right, sure enough, we get 900 million.
Let's try another problem. This time, you're going to fill in some of the steps along the way. I'm going to start this by writing each number in scientific notation. Once we have our scientific notation, we can rearrange our terms to group the numbers together, and the powers of 10 together. What would be the exponent for our power of 10? Enter your answer here.
Well, these are the 4 powers of 10 that are to be using or simplifying. Adding the exponents, I get 10 squared divided by 10 to the negative one. Now we subtract these exponents and I get 10 to the 3rd. Our 10 should have a power of
Now that we have our power of 10, let's simplify common factors in the numerator and the denominator. So what are the two common factors that you can simplify? Write the smaller factor in this box, and the larger factor in this one.
We can simplify as 7 10ths and then 8 10ths. Great thinking if you got those two. I knew this question might have been hard since we haven't focused on decimals in the course. If you think of this number as 24, and this number as go into 24 3 times. So 8 10ths goes in 2 and 4 10ths 3 times. We also know that multiply our numbers to get 15 20th times 10 to the 3rd. I know 15 20th is the same as 3 4th. Our final answer would be 7.5 times 10 to the second or 750.
For this last practice problem, I want you to write these four numbers in scientific notation. After you've written these in scientific notation, combine the powers of 10, and list that exponent here. Good luck.
If you've got these numbers for your scientific notations, and a positive 2 as your exponent, great work. These two numbers are large numbers, so they'll need positive exponents for the power of 10. And these numbers are less than 1, they're small, so they'll need negative exponents. Now I'm going to work with the powers of 10 down here to simplify them. When we multiply the same bases, we add the exponents. So we'll have 10 to the third, divided by 10 to the 1. And next, we divide the bases of 10, so we subtract the exponents, giving us 10 squared.
Now that we have our power of 10, let's try to simplify this first fraction part. What factors can divide into both the enumerator and the denominator? I want you to list the smaller factor or decimal here and the larger one here. Good luck.
We can remove a factor of 5 tenths and 7 tenths from the numerator and the denominator. 5 tenths divides into 1 and 5 tenths 3 times. And 5 tenths divides into 2 and 5 tenths 5 times. 7 tenths divides into 4 and 9 tenths 7 times. And divides into 2 and 1 tenth 3 times. If you're having trouble with the decimals, think about multiplying them all by 10. We know 5 goes into 15, 3 times. 5 goes into 25, 5 times. 7 would go into 21, 3 times. And 7 would go into 49, 7 times. We don't want to use this since it might confuse us with the powers of 10. But it's another way that might help you think about this division.
Now that we've simplified these numbers, let's finish our problem. Multiply the remaining factors and list them here, and then write our final answer as a fraction, the numberator and the denominator.
squared or 100. 35 times 100 is 3,500, and our denominator is just 9. This is our final fraction.