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| 16 Theory slides |
| 10 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
Decimal Form | Written as a Product or Division Expression | Scientific Notation |
---|---|---|
4505 | 4.505×1000 | 4.505×103 |
8320000 | 8.32×1000000 | 8.32×106 |
0.0005 | 5÷10000 | 5×10-4 |
0.0521 | 5.21÷100 | 5.21×10-2 |
move. Consider a number greater than 10. The decimal would move from right to left to make the number less than 10 but still greater than 1. The number of places the decimal moved indicates the positive exponent to be used for the base 10 power.
Ramsha and her friends did a research project about the highest mountains in the world. They found that Mount Everest is the world's highest mountain. It is located in the Himalayan mountain range. They also learned that Mount Elbrus is the highest peak of Europe. It is in southwestern Russia.
Ramsha and her friends continue to enjoy their school studies. Next is biology class! They use a microscope to examine two types of bacteria, Escherichia coli and Salmonella.
Rewrite the given expression in standard form if it is given in scientific notation. Or rewrite it in scientific notation if it is given in standard form.
Population | Rounded | |
---|---|---|
USA | 331002651 | 300000000 |
Brazil | 212559417 | 200000000 |
Turkey | 84339067 | 80000000 |
China | 1439323776 | 1000000000 |
Australia | 25499884 | 30000000 |
Population | Rounded | Scientific Notation | |
---|---|---|---|
USA | 331002651 | 300000000 | 3×108 |
Brazil | 212559417 | 200000000 | 2×108 |
Turkey | 84339067 | 80000000 | 8×107 |
China | 1439323776 | 1000000000 | 1×109 |
Australia | 25499884 | 30000000 | 3×107 |
Country | Population |
---|---|
USA | 3×108 |
Brazil | 2×108 |
Turkey | 8×107 |
China | 1×109 |
Australia | 3×107 |
Numbers written in scientific notation can be multiplied by using the properties of exponents. Two numbers written in scientific notation a×10b and c×10d can be multiplied by using the Commutative Property of Multiplication and the Product of Powers Property.
(a×10b)×(c×10d)=ac×10b+d
Remove parentheses
Commutative Property of Multiplication
Multiply
The result is 1.8×108. The first factor is greater than 1 and less than 10. The second factor is written as a power of 10. Therefore, the result is already in scientific notation.
Numbers written in scientific notation can be divided by using the properties of exponents. Use the Quotient of Powers Property to divide numbers written in scientific notation.
c×10da×10b=ca×10b−d
Write as a product of fractions
Calculate quotient
Ramsha's math teacher introduced how to multiply and divide numbers written in scientific notation. She then asked the class to work on the following examples.
Remove parentheses
Commutative Property of Multiplication
Multiply
am⋅an=am+n
Write as a product of fractions
Calculate quotient
anam=am−n
Ramsha loves reading science magazines. An article she read said that garden snails move at an incredibly slow speed of only 0.048 kilometers per hour!
Commutative Property of Multiplication
Multiply
am⋅an=am+n
Write as a fraction
Use a calculator
Round to 3 decimal place(s)
anam=am−n
a−(-b)=a+b
Add terms
Perform the following operation and write the result in scientific notation. If necessary, round the first factor of the result to one decimal.
Numbers written in scientific notation can be added or subtracted by adding or subtracting the first factors if the powers of 10 are the same.
(a×10b)±(c×10b)=(a±c)×10b
Ramsha's class took a field trip to learn about wind turbines. The turbines supply lots of households with electricity. A typical large wind turbine can produce 6×106 kilowatt-hour energy per year. A small wind turbine can produce 1.3×105 kilowatt-hour energy per year.
Perform the following operation and write the result in scientific notation. If necessary, round the first factor of the result to one decimal.
Yes, by using scientific notation.
Scientists are performing an experiment about bacteria. They note that about 3.6×108 bacteria grew in 4 petri dishes in total.
We know that about 3.6 * 10^8 bacteria are growing in 4 petri dishes in total. We are trying to find how many bacteria are growing in each petri dish. The number of bacteria in each petri dish is the same. That means can divide the total number of bacteria growing in 4 petri dishes by 4. 3.6*10^8/4 Let's first check whether both the numerator and denominator are in scientific notation. First Factor & Second Factor 3.6 ✓ & 10^8 ✓ 4 ✓ & ? Note that 3.6*10^8 is already in scientific notation. The number of petri dishes is less than 10 and greater than 1. However, we should express it as a product of 4 and a power of 10. We can do that by multiplying it by 10^0. That value equals 1 according to the Identity Property of Multiplication. 3.6*10^8/4*10^0 Great! Now, we can calculate the quotient in the expression using the Quotient of Powers Property.
Note that the result is not in scientific notation because 0.9 is less than 1. Let's move the decimal point one place to the right and decrease the power of 10. 0.9 * 10^8 ⇔ 9 * 10^7 Reconsider context of the problem. We can say that there are 9* 10^7 bacteria in each petri dishes.
This time we will find the total number of bacteria in 20 petri dishes. We will multiply the number of bacteria growing in each petri dish by the number of petri dishes. We already found that there are 9* 10^7 bacteria in each petri dishes in the Part I of the exercise. ( 9 * 10^7 ) * 20 Now we will need to evaluate the obtained expression. Let's start by checking whether both numbers are in scientific notation. First Factor & Second Factor 9 ✓ & 10^7 ✓ 20 ✓ & ? The number of petri dishes is not less than 10. That means it is not written in scientific notation. We can express it in scientific notation by rewriting 20 as 2* 10^1. ( 9 * 10^7 ) * ( 2* 10^1 ) Next, we can calculate the product in the expression by using the Product of Powers Property.
The first factor of 18 is not less than 10. It is not written in scientific notation. We can express it in scientific notation by moving the decimal point one place to the left and increase the power of 10. 18 * 10^8 ⇔ 1.8 * 10^9 A total of 1.8* 10^9 bacteria are growing in 20 petri dishes.
Sirius is the brightest star in Earth's night sky. It is about 8.6 light years from Earth. One light year is about 5.9×1012 miles. How far from Earth is Sirius measured in miles? Express the distance in scientific notation.
The distances between Sirius and Earth is given in light years. The distance of one light year in miles is also given. The question asks us to give the distance between Sirius and Earth in miles. We need to convert the distance given in light years to miles.
Let's start by multiplying the distance in light years by the number of miles in a light year. Distance in miles = ( 8.6 ) * ( 5.9 * 10^(12) ) Now we want to evaluate the obtained expression using the Product of Powers Property. Let's do it!
Note that 50.74 is not less than 10. That means our result is not written in scientific notation, yet. We can express it in scientific notation by moving the decimal point one place to the left and increasing the power of 10. 50.74 * 10^(12) ⇔ 5.074 * 10^(13) The distance between Sirius and Earth is about 5.074 * 10^(13) miles.
The mass of Earth is about 5.97×1024 kilograms. The mass of the Moon is about 7.35×1022 kilograms.
We want to find the combined mass of Earth and the Moon expressed in scientific notation. Let's begin by adding the mass of Earth to the mass of the Moon. ( 5.97 * 10^(24) )+( 7.35* 10^(22) ) The masses should be written with the same power of 10. With this in mind, let's rewrite the first number. The decimal point should be moved two places to the right and the power of 10 decreased by two units. 5.97 * 10^(24) ⇔ 597 * 10^(22) Continue to focus on the first factors of the numbers. These numbers are now like terms. That means they can be added. Factoring out 10^(22) gets the process started.
Notice that the result is not in scientific notation because the first factor is greater than 10. It needs to be rewritten. The decimal point moves two places to the left and the power of 10 increases by two units. 604.35 * 10^(22) ⇔ 6.0435 * 10^(24) We found the combined mass of Earth and the Moon equals 6.0435 * 10^(24) kilograms.
We are asked to find the difference between two distances. Subtracting the lesser distance from the greater one will accomplish that. Writing down the distances using the same power of 10 helps us compare them. Begin by rewriting 1.52 * 10^9. The decimal point moves one place to the right and the power of 10 is decreased by one unit. 1.52 * 10^9 ⇔ 15.2 * 10^8 Now the second factors of the products are the same. We can now compare the greatest distance between the Sun and Earth with the greatest distance between the Sun and Saturn. 1.52 * 10^8 < 15.2 * 10^8 Let's express their difference as an expression. ( 15.2 * 10^8 ) - ( 1.52 * 10^8 ) The numbers are written with the same power of 10. That allows for us to factor out 10^8.
We end up with 13.68 which is greater than 10. The result is not written in scientific notation yet. Moving the decimal point one place to the left makes the first factor less than 10. That results in increasing the power of 10 by one unit. 13.68 * 10^8 ⇔ 1.368 * 10^9 The difference between the given two distances equals 1.368* 10^9 kilometers.