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Scientists and astronomers make calculations using huge and tiny numbers. They could go all the way up to

100000000000

or as low as

0.00000000001 .

Writing these numbers can take a lot of time while making calculations. A more practical way of writing numbers exists called scientific notation. This concept will be explored in this lesson.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Associative Property of Multiplication
Commutative Property of Multiplication
Product of Powers Property
Quotient of Powers Property
Negative Exponent
Like Terms

Challenge

Distance From the Earth to the Sun

The distance from the Earth to the Sun is about

150000000

kilometers.

An astronomer wants to find out how long a spaceship would take to fly to the Sun. He will need to write the given distance several times to make this calculation. Is there a way to make the given number shorter? If yes, how?

Discussion

Scientific Notation

Scientific notation is a compact way of writing very large or very small numbers. A number written in scientific notation is expressed as a product of two numbers.

a \times 1 0^{b}

In this form, the first factor is greater than or equal to

1

and less than

10 .

In other words, it needs to be in the interval

[1, 10) .

The second factor is a power of

10

where

b

is an integer. For example, the number

4

million can be rewritten as the product of

4

and a multiple of

10 .

Then, the multiple of

10

is rewritten as a base

10

power.

4000000 = 4 \times 1000000 = 4 \times 1 0^{6}

Very small decimal numbers can also be written in scientific notation. Consider a number where there are many zeros before the significant figures. Take as

0.000342

as an example.

0.000342 = 3.42 \div 10000 = 3.42 \times 1 0^{- 4}

In such cases, numbers are expressed as a division by a multiple of

10 .

Division by a multiple of

10

is equivalent to multiplication by a base

10

power with a negative exponent. Consider a few more examples of numbers written in scientific notation.

Decimal Form	Written as a Product or Division Expression	Scientific Notation
$4505$	$4.505 \times 1000$	$4.505 \times 1 0^{3}$
$8320000$	$8.32 \times 1000000$	$8.32 \times 1 0^{6}$
$0.0005$	$5 \div 10000$	$5 \times 1 0^{- 4}$
$0.0521$	$5.21 \div 100$	$5.21 \times 1 0^{- 2}$

Intuitive Method: Rewriting a Number in Scientific Notation

An intuitive method to rewrite a number into scientific notation is to count the number of places the decimal needs to 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.

Similarly, for numbers less than

1,

such as

0.000022,

the decimal will move from left to right to make the number greater than or equal to

1

and less than

10 .

In this case, the number of places moved indicates the negative exponent to be used for the base

10

power.

Scientific notation is not only a convenient way to express cumbersome numbers. It also eases the comparison of numerical order of magnitude. For example, it may be difficult to determine how much larger

237000000

is compared to

4530000 .

However, it is easier to see that

2.37 \times 1 0^{8}

and

4.5 \times 1 0^{6}

differ by a factor of about

1 0^{2} = 100 .

Example

Rewriting the Heights of Mountains

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.

a The peak of Mount Everest is at

8848

meters above the sea level. Rewrite this height in scientific notation.

b Mount Elbrus is an extinct volcano with twin cones that reach

1.851 \times 1 0^{4}

feet. Rewrite this height in standard form.

Hint

a Start by placing the decimal point after the first non-zero digit.

b The power of

10

is positive. That means the decimal point moves to the right.

Solution

a The height of Mount Everest is

8848

meters. Notice that this number is four digits long when written in standard form. Start by placing the decimal point after the first non-zero digit to write it in scientific notation.

8.848 ↑ First nonzero digit

Next, count the digits after the decimal point.

8.848 ↑ Digits after the decimal point

There are

three

digits after the decimal point. This number will be written as the exponent of

10 .

Standard Form 8848 Scientific Notation 8.848 \times 1 0^{3}

b This time the number is given in the scientific notation and the task is to rewrite it into standard form.

1.851 \times 1 0^{4}

The power of

10

4 .

It has a positive exponent. Therefore, the decimal point will be moved to the right

4

times. In other words, the first factor

1.851

will be multiplied by

10

four

times.

The diagram shows that the decimal point continues to move after the nonzero digits. Zeroes can be added to the end of the number until the move is done. Note that adding zeroes to the end of a decimal number after the decimal point does not change the value of the number.

1.851 = 1.8510000 \dots

In this case, there are already three nonzero digits. Only one zero needs to be added to the end of the number. Now, the number is in standard form as a five-digit number.

Scientific Notation 1.851 \times 1 0^{4} Standard Form 18510

Example

Rewriting the Lengths of Bacteria

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.

a The length of an E. coli bacterium is two micrometers long. That is equal to

0.000002

meters. Write this length in scientific notation in meters.

b The length of a salmonella bacterium is

1.5 \times 1 0^{- 6}

meters. Rewrite this length in standard form.

Hint

a Start by placing the decimal point after the first non-zero digit.

b The power of

10

is negative. That means the decimal point moves to the left.

Solution

a The length of an Escherichia coli bacterium is given as

0.000002

meters. The goal is to write this number in scientific notation. Begin by placing the decimal point after the first non-zero digit.

0.000002 . ↑ First nonzero digit

Then, determine the power of

10 .

That is done by counting the number of digits before the new decimal point.

0.000002 . ↑ Digits before the new decimal point

There are

six

digits before the decimal point. The given number

0.000002

is less than

1 .

That means the exponent will be negative.

Standard Form 0.000002 Scientific Notation 2 \times 1 0^{- 6}

b The length of a salmonella bacterium is given in scientific notation.

1.5 \times 1 0^{- 6}

Notice that the power of

10

is negative. This means that the decimal point will be moved to the

left six

times.

A zero is added every time the decimal moves to the left of the given digits. Remember to write an additional zero before the decimal point. Finally, the number is rewritten in standard form as an eight-digit number.

Scientific Notation 1.5 \times 1 0^{- 6} Standard Form 0.0000015

Pop Quiz

Translating Between Scientific Notation and Standard Form

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.

Example

Examining the Populations of Some Countries

Ramsha's school has an interactive electronic map that shows population data around the world! Ramsha's geography teacher asked her class to examine the populations of some countries.

world map showing population of some countries

They used this map to see the numbers of people living in Brazil, Turkey, China, Australia, and the United States of America.

a Write the number of people living in these countries in scientific notation by rounding the given numbers to the greatest place value.

b Sort the countries from greatest to least population.

Hint

a Round the numbers to the greatest place value. Then count the zeros.

b Examine the powers of

10

for each country. Then, compare the first factors of the numbers with the same power of

10 .

Solution

a The populations of the five countries will be written in scientific notation one at a time. Start by rounding all the numbers to the greatest place value in a table.

	Population	Rounded
USA	$331002651$	$300000000$
Brazil	$212559417$	$200000000$
Turkey	$84339067$	$80000000$
China	$1439323776$	$1000000000$
Australia	$25499884$	$30000000$

Now all the rounded numbers can be rewritten as a single digit times a power of

10 .

Count the zeros to determine the power of

10

for each number. For instance, the rounded population of the USA can be rewritten in this way.

300000000 = 3 \times 1 0^{8}

Apply the same method so the rounded populations of other countries in standard form can be rewritten in scientific notation.

	Population	Rounded	Scientific Notation
USA	$331002651$	$300000000$	$3 \times 1 0^{8}$
Brazil	$212559417$	$200000000$	$2 \times 1 0^{8}$
Turkey	$84339067$	$80000000$	$8 \times 1 0^{7}$
China	$1439323776$	$1000000000$	$1 \times 1 0^{9}$
Australia	$25499884$	$30000000$	$3 \times 1 0^{7}$

b It is time to determine which country has the greatest population! This process requires comparing the numbers written in scientific notation from Part A.

Country	Population
USA	$3 \times 1 0^{8}$
Brazil	$2 \times 1 0^{8}$
Turkey	$8 \times 1 0^{7}$
China	$1 \times 1 0^{9}$
Australia	$3 \times 1 0^{7}$

Examine the powers of

10 .

Begin with identifying the largest power. China has the largest. That means it has the greatest population. Now compare the numbers with the same power of

10 .

Start with

1 0^{8}

because

1 0^{8}

is greater than

1 0^{7} .

USA 3 \times 1 0^{8} Brazil 2 \times 1 0^{8}

The powers of

10

for the US and Brazil are equal. That means their first factors should be compared. The value

3

is larger than

2 .

That means the population of the USA is greater than Brazil. Next, compare the populations of Turkey and Australia.

Turkey 8 \times 1 0^{7} Australia 3 \times 1 0^{7}

Again, the powers of

10

are the same. This indicates that the values of

8

and

3

should be checked. Well,

8

is greater than

3 .

That means the population of Turkey is greater than Australia. Now the countries can be sorted from greatest to least population.

China USA Brazil Turkey Australia 1 \times 1 0^{9} 3 \times 1 0^{8} 2 \times 1 0^{8} 8 \times 1 0^{7} 3 \times 1 0^{7}

Discussion

Multiplying Numbers in Scientific Notation

Numbers written in scientific notation can be multiplied by using the properties of exponents. Two numbers written in scientific notation $a \times 1 0^{b}$ and $c \times 1 0^{d}$ can be multiplied by using the Commutative Property of Multiplication and the Product of Powers Property.

$(a \times 1 0^{b}) \times (c \times 1 0^{d}) = a c \times 1 0^{b + d}$

The first factors of the numbers are multiplied like integers or decimal numbers. Then, the exponents of the second factors are added. Since they have the same base

10,

the Product of Powers Property can be used. As an example, consider the following product.

(1.5 \times 1 0^{2}) \times (12 \times 1 0^{5})

Three steps can be followed to multiply these numbers.

Write Each Number in Scientific Notation

It is helpful to check whether both of the numbers are written in scientific notation. Recall that a number is in scientific notation if it is written as a product of two numbers. One of these numbers should be a power of

10 .

The other number should be greater than or equal to

1

and less than

10 .

First Factor 1.5 12 \times \times \times Second Factor 1 0^{2} 1 0^{5} ✓ \times

The first number is in scientific notation. The second number is not. Its first factor

12

is greater than

10 .

Note that

12

can also be expressed as

12.0 .

The decimal point needs to move one unit to the left for the first factor to become less than

10 .

The second factor's power of

10

is then increased by that number of moves,

1 .

12 \times 1 0^{5} \Leftrightarrow 1.2 \times 1 0^{5 + 1} \Leftrightarrow 1.2 \times 1 0^{6} ✓

Multiply the First Factors

Both numbers are now written in scientific notation. The first factors of both numbers can be multiplied by using the Commutative Property of Multiplication.

(1.5 \times 1 0^{2}) \times (1.2 \times 1 0^{6})

RemovePar

Remove parentheses

1.5 \times 1 0^{2} \times 1.2 \times 1 0^{6}

CommutativePropMult

Commutative Property of Multiplication

1.5 \times 1.2 \times 1 0^{2} \times 1 0^{6}

Multiply

1.8 \times 1 0^{2} \times 1 0^{6}

Multiply the Second Factors

Next, the powers with base

10

are multiplied. Remember to use the Product of Powers Property — exponents are added when multiplying powers with the same base.

1.8 \times 1 0^{2} \times 1 0^{6}

MultPow

$a^{m} \cdot a^{n} = a^{m + n}$

1.8 \times 1 0^{8}

Write the Result in Scientific Notation

The result is $1.8 \times 1 0^{8} .$ 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.

Discussion

Dividing Numbers 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.

$\frac{a \times 1 0 ^{b}}{c \times 1 0 ^{d}} = \frac{a}{c} \times 1 0^{b - d}$

Consider the division of the following two numbers.

\frac{0 . 3 6 \times 1 0 ^{23}}{1 2 0 0}

These numbers written in scientific notation, and they can be divided in four steps. It is similar to the process of multiplying numbers written in scientific notation.

Write Each Number in Scientific Notation

Start by checking whether each number is written in scientific notation.

First Factor 0.36 \times \times Second Factor 1 0^{23} \times

The first factor is less than

1 .

That means the dividend still needs to be written in scientific notation. Move the decimal point to the right until

0.36

is greater than

1

and less than

10 .

Then, decrease the second factor's exponent by the number of moves.

0.36 \times 1 0^{23} \Leftrightarrow 3.6 \times 1 0^{22}

Next, consider the divisor. The divisor

1200

is a whole number. Note that

1200

can also be expressed as

1200.0 .

Now, move the decimal point three units to the left to get a first factor that is less than

10

and greater than

1 .

The second factor can then be written as

10

to the power of

3 .

1200 \Leftrightarrow 1.2 \times 1 0^{3}

Divide the First Factors

Now that both the dividend and divisor are written in scientific notation, the first factors of both can be divided.

\frac{3 . 6 \times 1 0 ^{22}}{1 . 2 \times 1 0 ^{3}}

WriteProdFrac

Write as a product of fractions

\frac{3 . 6}{1 . 2} \times \frac{1 0 ^{22}}{1 0 ^{3}}

CalcQuot

Calculate quotient

3 \times \frac{1 0 ^{22}}{1 0 ^{3}}

Divide the Second Factors

Next, use the Quotient of Powers Property to divide the second factors. This property states that when dividing powers with the same base, the exponents are subtracted.

3 \times \frac{1 0 ^{22}}{1 0 ^{3}}

DivPow

$\frac{a ^{m}}{a ^{n}} = a^{m - n}$

3 \times 1 0^{19}

Write the Result in Scientific Notation

Finally, write the resulting number in scientific notation. Notice that in this instance, the quotient is already in scientific notation.

3 \times 1 0^{19}

Example

Finding the Product and 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.

A girl is standing in front of a board. On the left, Example I shows a multiplication written in scientific notation. On the right, Example II shows a division in scientific notation.

a Solve along with Ramsha to find the product given in Example I. Express the result in scientific notation.

b Solve along with Ramsha to find the quotient given in Example II. Express the result in scientific notation.

Hint

a Use the Commutative Property of Multiplication and the Product of Powers Property.

b Use the Quotient of Powers Property.

Solution

a Ramsha is finding the product of two numbers. It is important to check that both numbers are in scientific notation before she finds their product. She can do this by checking for two certain characteristics.

The first factor must be greater than or equal to $1$ and less than $10 .$
The second factor must be a power of $10 .$

It is helpful to organize this check in the following way.

First Factor 1.4 5 \times \times Second Factor 1 0^{44} 1 0^{14} ✓ ✓

Both numbers meet the two characteristics. They are written in scientific notation. Now their first factors can be multiplied by using the Commutative Property of Multiplication. Their second factors can be multiplied by using the Product of Powers Property.

(1.4 \times 1 0^{44}) \times (5 \times 1 0^{14})

RemovePar

Remove parentheses

1.4 \times 1 0^{44} \times 5 \times 1 0^{14}

CommutativePropMult

Commutative Property of Multiplication

1.4 \times 5 \times 1 0^{44} \times 1 0^{14}

Multiply

7 \times 1 0^{44} \times 1 0^{14}

MultPow

$a^{m} \cdot a^{n} = a^{m + n}$

7 \times 1 0^{58}

The product of the multiplication is already in scientific notation!

b Notice that Example II uses the same numbers as Example I. It was found in Part A that both the dividend and divisor are already written in scientific notation.

First Factor 1.4 5 \times \times Second Factor 1 0^{44} 1 0^{14} ✓ ✓

Recall how to divide numbers written in scientific notation. The first factors are divided like fractions and the second factors are divided by using Quotient of Powers Property.

\frac{1 . 4 \times 1 0 ^{44}}{5 \times 1 0 ^{14}}

WriteProdFrac

Write as a product of fractions

\frac{1 . 4}{5} \times \frac{1 0 ^{44}}{1 0 ^{14}}

CalcQuot

Calculate quotient

0.28 \times \frac{1 0 ^{44}}{1 0 ^{14}}

DivPow

$\frac{a ^{m}}{a ^{n}} = a^{m - n}$

0.28 \times 1 0^{30}

Notice that the result is not in scientific notation yet because the first factor is less than

1 .

The decimal point needs to be moved one unit to the right. That move means the power of

10

must be decreased by one. Now it is in scientific notation.

0.28 \times 1 0^{30} \Leftrightarrow 2.8 \times 1 0^{29}

Example

Snails on the Way

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!

a How many kilometers can a snail go if it is on the move for a full day of

24

hours? Perform the needed operations in scientific notation. Also express the result in scientific notation. Round the result to two decimal places if necessary.

b How long does it take a snail to move one meter? Perform the needed operations in scientific notation. Also express the result in scientific notation. Round the result to two decimal places if necessary.

Hint

a Distance traveled can be calculated by multiplying the speed by the time.

b Use the Quotient of Powers Property.

Solution

a It is given that a snail can move

0.048

kilometers in one hour. One day is

24

hours, so the distance traveled in one day can be found by multiplying the speed of the snail by

24 .

Distance = Speed \times Time ⇓ Distance = 0.048 \times 24

Begin by rewriting both numbers in scientific notation. Notice that

0.048

is less than

1 .

The decimal point moves two units to the right to become greater than

1 .

Next, the two unit move to the right means the base

10

power will have an exponent of

- 2

0.048 \times 1 0^{0} = 4.8 \times 1 0^{- 2}

Now consider

24 .

A number greater than

10

moves a certain number of units to the left. This case requires a one unit move to the left. Recall that movement to the left means the exponent of the base

10

power will be positive. The second factor is

1 0^{1} .

24 \times 1 0^{0} = 2.4 \times 1 0^{1}

Now that both numbers are in scientific notation, multiply them by using Commutative Property of Multiplication and Product of Powers Property.

(4.8 \times 1 0^{- 2}) \times (2.4 \times 1 0^{1})

CommutativePropMult

Commutative Property of Multiplication

(4.8 \times 2.4) \times (1 0^{- 2} \times 1 0^{1})

Multiply

11.52 \times (1 0^{- 2} \times 1 0^{1})

MultPow

$a^{m} \cdot a^{n} = a^{m + n}$

11.52 \times 1 0^{- 1}

The result is not in scientific notation. It will need to be rewritten. Follow the same method as done previously. Move the decimal point one unit to the left and increase the power of

10

by that same value,

1 .

11.52 \times 1 0^{- 1} = 1.152 \times 1 0^{0}

Finally, round the obtained result to two decimals.

1.152 \times 1 0^{0} \approx 1.15 \times 1 0^{0}

In a full day, the snail can move about

1.15 \times 1 0^{0}

1.15

kilometers. Quite impressive!

b This time it is asked for the time that it takes to move one meter for the snail. Since the speed of the snail is given in terms of kilometers per hour, at first rewrite one meter in terms of kilometers. There are

1000

meters in

1

kilometer, so

1

meter is one-thousands or

0.001

of a kilometer.

1 meter = 0.001 kilometers

Now, rearrange the distance formula to use it for the time by dividing both sides of the equation by speed.

Distance = Speed \times Time ⇓ Time = \frac{Distance}{Speed}

Before using the obtained formula, rewrite the numbers to have them in scientific notation. To do so, move the decimal points to the right to make the first factors greater than or equal to

1

and decrease the powers of

10

according to amount of decimal points moved.

Then, perform the division to find the time that the snail needs to move

1 \times 1 0^{- 3}

kilometers.

\frac{1 \times 1 0 ^{- 3}}{4 . 8 \times 1 0 ^{- 2}}

WriteAsFrac

Write as a fraction

\frac{1}{4 . 8} \times \frac{1 0 ^{- 3}}{1 0 ^{- 2}}

UseCalc

Use a calculator

(0.208333 \dots) \times \frac{1 0 ^{- 3}}{1 0 ^{- 2}}

RoundDec

Round to $3$ decimal place(s)

0.208 \times \frac{1 0 ^{- 3}}{1 0 ^{- 2}}

DivPow

$\frac{a ^{m}}{a ^{n}} = a^{m - n}$

0.208 \times 1 0^{- 3 - (- 2)}

SubNeg

$a - (- b) = a + b$

0.208 \times 1 0^{- 3 + 2}

AddTerms

Add terms

0.208 \times 1 0^{- 1}

Since the first factor of the obtained result is less than

1,

move the decimal point one unit to the right and decrease the power of

10

one unit as well to have it in scientific notation.

0.208 \times 1 0^{- 1} = 2.08 \times 1 0^{- 2}

This means that the snail moves one meter in about

2.08 \times 1 0^{- 2}

hours. For those curious, this is about

75

seconds. Turns out snails are actually not that slow!

Pop Quiz

Multiply or Divide Numbers in Scientific Notation

Perform the following operation and write the result in scientific notation. If necessary, round the first factor of the result to one decimal.

Discussion

Add and Subtract Numbers in Scientific Notation

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.

$(a \times 1 0^{b}) \pm (c \times 1 0^{b}) = (a \pm c) \times 1 0^{b}$

Recall that rewriting the result in scientific notation is necessary when the first factor is greater than

10

or less than

1 .

In such cases, the exponent of

10

is increased or decreased by moving the decimal point. Consider the following addition example.

(3 \times 1 0^{12}) + (0.15 \times 1 0^{15})

Adding these numbers calls for three steps to be followed.

Change the Exponents of

10

to Be the Same

Two options can be considered. Either increase the power of

1 0^{12}

or decrease the power of

1 0^{15} .

The second option will be considered here. Move the decimal point to the right three units. Then decrease the power of

10

by the same number of moves, three units.

0.15 \times 1 0^{15} \Leftrightarrow 150 \times 1 0^{12}

Now, both of the numbers have the same

10 -

base factor with the exponent of

12 .

Add or Subtract the First Factors

The obtained numbers are now like terms. That means the first factors are ready to be added.

(3 \times 1 0^{12}) + (150 \times 1 0^{12})

FactorOut

Factor out $1 0^{12}$

(3 + 150) \times 1 0^{12}

AddTerms

Add terms

153 \times 1 0^{12}

Write the Result in Scientific Notation

Notice that the result

153 \times 1 0^{12}

is not in scientific notation. The first factor

153

is greater than

10 .

It needs to be less than

10

and greater than or equal to

1 .

That is done by moving the decimal point two units to the left. Then increase the power of

10

by two units.

153 \times 1 0^{12} \Leftrightarrow 1.53 \times 1 0^{14}

Subtracting numbers in scientific notation follows the same process. Note that writing these numbers in standard form is another way to add or subtract them.

Example

Calculating the Energy Produced by Wind Turbines

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 $6 \times 1 0^{6}$ kilowatt-hour energy per year. A small wind turbine can produce $1.3 \times 1 0^{5}$ kilowatt-hour energy per year.

a How much energy can a typical large and small wind turbine produce together in one year? Write the result in scientific notation.

b What is the difference in the produced energies of the two wind turbines per year in scientific notation?

Hint

a Rewrite the numbers to have the same power of

10 .

When the numbers are like terms, add the first factors of these numbers.

b Move the decimal point to rewrite the numbers. When the numbers are like terms, subtract the first factors of these numbers.

Solution

a The goal is to find the total energy that is produced by a typical large wind turbine and a small wind turbine together. The given energy amounts per year will be added to find that number.

(6 \times 1 0^{6}) + (1.3 \times 1 0^{5})

The powers of

10

need to be the same to be able to add these numbers. The first number can be rewritten by moving the decimal point one unit to the right. That results in its power of

10

decreasing by one unit.

6 \times 1 0^{6} \Leftrightarrow 60 \times 1 0^{5}

The first factors of the numbers can be added since these numbers are now like terms.

60 \times 1 0^{5} + 1.3 \times 1 0^{5}

FactorOut

Factor out $1 0^{5}$

(60 + 1.3) \times 1 0^{5}

AddTerms

Add terms

61.3 \times 1 0^{5}

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 can be moved one unit to the left. That results in its power of

10

increasing by one.

61.3 \times 1 0^{5} \Leftrightarrow 6.13 \times 1 0^{6}

In total, a typical large and small wind turbine produce

6.13 \times 1 0^{6}

kilowatt-hour energy per year. That is enough to meet the electricity demand of around

1600

average households per year.

b This time the difference between the energy amounts will be found.

(6 \times 1 0^{6}) - (1.3 \times 1 0^{5})

Perform this operation by setting the powers of

10

as the same. Recall that the first number was already rewritten in Part A.

(6 \times 1 0^{6}) (60 \times 1 0^{5}) - ⇓ - (1.3 \times 1 0^{5}) (1.3 \times 1 0^{5})

Subtraction can be performed now that they are like terms.

(60 \times 1 0^{5}) - (1.3 \times 1 0^{5})

FactorOut

Factor out $1 0^{5}$

(60 - 1.3) \times 1 0^{5}

SubTerms

Subtract terms

58.7 \times 1 0^{5}

The result is not in scientific notation. It can be rewritten by moving the decimal point one unit to the left. That results in the power of

10

increasing by one unit.

58.7 \times 1 0^{5} \Leftrightarrow 5.87 \times 1 0^{6}

The difference in produced energies of the two type of wind turbines is

5.87 \times 1 0^{6}

kilowatt-hour per year.

Pop Quiz

Add or Subtract Numbers in Scientific Notation

Perform the following operation and write the result in scientific notation. If necessary, round the first factor of the result to one decimal.

Closure

Writing the Distance From the Earth to the Sun

The initial challenge of this collection stated that the distance from the Earth to the Sun is about

150000000

kilometers.

An astronomer making calculations with this number needs to write it several times to find the time it would take a spaceship to fly to the Sun. Is there a way to make this notation shorter? If yes, how?

Answer

Yes, by using scientific notation.

Hint

Is there a way to write the given number as a product of two numbers where one of them is a power of $10 ?$

Solution

This number can be written as a product of two numbers to express it in shorter notation.

150000000 = First Factor \times Second Factor

There are lots zeros at the end of the given number — seven zeros to be exact. These seven zeros can be written as a power of

10 .

The remaining two digits

1

and

5

can be written as the other factor.

150000000 ⇓ 15 \times 1 0^{7}

Consider the main topic of this lesson, scientific notation. A number written in scientific notation needs to have the following two certain characteristics.

The first factor must be greater than or equal to $1$ and less than $10 .$
The second factor must be a power of $10 .$

With this in mind, consider the obtained number in the quest to represent

150000000

kilometers.

15 \times 1 0^{7}

The first factor is greater than

10 .

That means it needs to be rewritten once again. That can be done by moving the decimal point one unit to the left. This results in the power of

10

increasing by one unit.

15 \times 1 0^{7} \Leftrightarrow 1.5 \times 1 0^{8}

Writing it this way means the astronomer spends less time writing out

150000000

kilometers multiple times. That might not sound like such a big deal, but over time it really makes a difference. Imagine writing such a large number over and over again. Thank you scientific notation!

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Catch-Up and Review

Intuitive Method: Rewriting a Number in Scientific Notation

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