scientific notation problem solving questions

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math tutorials for students majoring in the earth sciences

Scientific Notation - Practice Problems

Solving earth science problems with scientific notation, × div[id^='image-'] {position:static}div[id^='image-'] div.hover{position:static} introductory problems.

These problems cover the fundamentals of writing scientific notation and using it to understand relative size of values and scientific prefixes.

Problem 1: The distance to the moon is 238,900 miles. Write this value in scientific notation.

Problem 2: One mile is 1609.34 meters. What is the distance to the moon in meters using scientific notation?

`1609.34 m/(mi) xx 238","900 mi` = 384,400,000 m

Notice in the above unit conversion the 'mi' units cancel each other out because 'mi' is in the denominator for the first term and the numerator for the second term

Problem 4: The atomic radius of a magnesium atom is approximately 1.6 angstroms, which is equal to 1.6 x 10 -10 meters (m). How do you write this length in standard form?

0.00000000016 m

Fissure A = 40,0000 m Fissure B = 5,0000 m

This shows fissure A is larger (by almost 10 times!). The shortcut to answer a question like this is to look at the exponent. If both coefficients are between 1-10, then the value with the larger exponent is the larger number.

Problem 6: The amount of carbon in the atmosphere is 750 petagrams (pg). One petagram equals 1 x 10 15 grams (g). Write out the amount of carbon in the atmosphere in (i) scientific notation and (ii) standard decimal format.

The exponent is a positive number, so the decimal will move to the right in the next step.

750,000,000,000,000,000 g

Advanced Problems

Scientific notation is used in solving these earth and space science problems and they are provided to you as an example. Be forewarned that these problems move beyond this module and require some facility with unit conversions, rearranging equations, and algebraic rules for multiplying and dividing exponents. If you can solve these, you've mastered scientific notation!

Problem 7: Calculate the volume of water (in cubic meters and in liters) falling on a 10,000 km 2 watershed from 5 cm of rainfall.

`10,000 km^2 = 1 xx 10^4 km^2`

5 cm of rainfall = `5 xx 10^0 cm`

Let's start with meters as the common unit and convert to liters later. There are 1 x 10 3 m in a km and area is km x km (km 2 ), therefore you need to convert from km to m twice:

`1 xx 10^3 m/(km) * 1 xx 10^3 m/(km) = 1 xx 10^6 m^2/(km)^2` `1 xx 10 m^2/(km)^2 * 1 xx 10^4 km^2 = 1 xx 10^10 m^2` for the area of the watershed.

For the amount of rainfall, you should convert from centimeters to meters:

`5 cm * (1 m)/(100 cm)= 5 xx 10^-2 m`

`V = A * d`

When multiplying terms with exponents, you can multiply the coefficients and add the exponents:

`V = 1 xx 10^10 m^2 * 5.08 xx 10^(-2) m = 5.08 xx 10^8 m^3`

Given that there are 1 x 10 3 liters in a cubic meter we can make the following conversion:

`1 xx 10^3 L * 5.08 xx 10^8 m^3 = 5.08 xx 10^11 L`

Step 5. Check your units and your answer - do they make sense?

`V = 4/3 pi r^3`

Using this equation, plug in the radius (r) of the dust grains.

`V = 4/3 pi (2 xx10^(-6))^3m^3`

Notice the (-6) exponent is cubed. When you take an exponent to an exponent, you need to multiply the two terms

`V = 4/3 pi (8 xx10^(-18)m^3)`

Then, multiple the cubed radius times pi and 4/3

`V = 3.35 xx 10^(-17) m^3`

`m = 3.35 xx 10^(-17) m^3 * 3300 (kg)/m^3`

Notice in the equation above that the m 3 terms cancel each other out and you are left with kg

`m = 1.1 xx 10^(-13) kg`

`V = 4/3 pi (2.325 xx10^(15) m)^3`

`V = 5.26 xx10^(46) m^3`

Number of dust grains = `5.26 xx10^(46) m^3 xx 0.001` grains/m 3

Number of dust grains = `5.26xx10^43 "grains"`

Total mass = `1.1xx10^(-13) (kg)/("grains") * 5.26xx10^43 "grains"`

Notice in the equation above the 'grains' terms cancel each other out and you are left with kg

Total mass = `5.79xx10^30 kg`

If you feel comfortable with this topic, you can go on to the assessment . Or you can go back to the Scientific Notation explanation page .

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4.4: Scientific Notation

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Learning Objectives

Define decimal and scientific notation

Convert between scientific and decimal notation

Multiply and divide numbers expressed in scientific notation
Solve application problems involving scientific notation

Before we can convert between scientific and decimal notation, we need to know the difference between the two. S cientific notation is used by scientists, mathematicians, and engineers when they are working with very large or very small numbers. Using exponential notation, large and small numbers can be written in a way that is easier to read.

When a number is written in scientific notation, the exponent tells you if the term is a large or a small number. A positive exponent indicates a large number and a negative exponent indicates a small number that is between 0 and 1. It is difficult to understand just how big a billion or a trillion is. Here is a way to help you think about it.

1 billion can be written as 1,000,000,000 or represented as $2\times10^9$.

Outlined in the box below are some important conventions of scientific notation format.

Scientific Notation

A positive number is written in scientific notation if it is written as $1\leq{a}<10$, and n is an integer.

Look at the numbers below. Which of the numbers is written in scientific notation?

Now let’s compare some numbers expressed in both scientific notation and standard decimal notation in order to understand how to convert from one form to the other. Take a look at the tables below. Pay close attention to the exponent in the scientific notation and the position of the decimal point in the decimal notation.

Convert from decimal notation to scientific notation

To write a large number in scientific notation, move the decimal point to the left to obtain a number between 1 and 10. Since moving the decimal point changes the value, you have to multiply the decimal by a power of 10 so that the expression has the same value.

Let’s look at an example.

$\begin{array}{r}180,000.=18,000.0\times10^{1}\\1,800.00\times10^{2}\\180.000\times10^{3}\\18.0000\times10^{4}\\1.80000\times10^{5}\\180,000=1.8\times10^{5}\end{array}$

Notice that the decimal point was moved 5 places to the left, and the exponent is 5.

Write the following numbers in scientific notation.

$920,000,000$
$10,200,000$
$100,000,000,000$

[reveal-answer q=”628″]Show Solution[/reveal-answer] [hidden-answer a=”628″]

$\underset{\longleftarrow}{920,000,000}=920,000,000.0$, move the decimal point 8 times to the left and you will have $9.2\times10^{8}$
$\underset{\longleftarrow}{10,200,000}=10,200,000.0=1.02\times10^{7}$, note here how we included the 0 and the 2 after the decimal point. In some disciplines, you may learn about when to include both of these. Follow instructions from your teacher on rounding rules.
$\underset{\longleftarrow}{100,000,000,000}=100,000,000,000.0=1.0\times10^{11}$

[/hidden-answer]

To write a small number (between 0 and 1) in scientific notation, you move the decimal to the right and the exponent will have to be negative, as in the following example.

$\begin{array}{r}\underset{\longrightarrow}{0.00004}=00.0004\times10^{-1}\\000.004\times10^{-2}\\0000.04\times10^{-3}\\00000.4\times10^{-4}\\000004.\times10^{-5}\\0.00004=4\times10^{-5}\end{array}$

You may notice that the decimal point was moved five places to the right until you got to the number 4, which is between 1 and 10. The exponent is $−5$.

$0.0000000000035$
$0.0000000102$
$0.00000000000000793$

[reveal-answer q=”229054″]Show Solution[/reveal-answer] [hidden-answer a=”229054″]

$\underset{\longrightarrow}{0.0000000000035}=3.5\times10^{-12}$, we moved the decimal 12 times to get to a number between 1 and 10
$\underset{\longrightarrow}{0.0000000102}=1.02\times10^{-8}$
$\underset{\longrightarrow}{0.00000000000000793}=7.93\times10^{-15}$

In the following video you are provided with examples of how to convert both a large and a small number in decimal notation to scientific notation.

$Thumbnail for the embedded element "Examples: Write a Number in Scientific Notation"$

A YouTube element has been excluded from this version of the text. You can view it online here: pb.libretexts.org/ba/?p=94

Convert from scientific notation to decimal notation

You can also write scientific notation as decimal notation. Recall the number of miles that light travels in a year is $5\times10^{-8}$ mm. To write each of these numbers in decimal notation, you move the decimal point the same number of places as the exponent. If the exponent is positive , move the decimal point to the right. If the exponent is negative , move the decimal point to the left.

$\begin{array}{l}5.88\times10^{12}=\underset{\longrightarrow}{5.880000000000.}=5,880,000,000,000\\5\times10^{-8}=\underset{\longleftarrow}{0.00000005.}=0.00000005\end{array}$

For each power of 10, you move the decimal point one place. Be careful here and don’t get carried away with the zeros—the number of zeros after the decimal point will always be 1 less than the exponent because it takes one power of 10 to shift that first number to the left of the decimal.

Write the following in decimal notation.

$4.8\times10{-4}$
$3.08\times10^{6}$

[reveal-answer q=”489774″]Show Solution[/reveal-answer] [hidden-answer a=”489774″]

$\underset{\longleftarrow}{4.8\times10^{-4}}=\underset{\longleftarrow}{.00048}$
$\underset{\longrightarrow}{3.08\times10^{6}}=\underset{\longrightarrow}{3080000}$

Think About It

To help you get a sense of the relationship between the sign of the exponent and the relative size of a number written in scientific notation, answer the following questions. You can use the textbox to wirte your ideas before you reveal the solution.

1. You are writing a number that is greater than 1 in scientific notation. Will your exponent be positive or negative?

[practice-area rows=”1″][/practice-area]

2. You are writing a number that is between 0 and 1 in scientific notation. Will your exponent be positive or negative?

3. What power do you need to put on 10 to get a result of 1?

[practice-area rows=”1″][/practice-area] [reveal-answer q=”824936″]Show Solution[/reveal-answer] [hidden-answer a=”824936″] 1. You are writing a number that is greater than 1 in scientific notation. Will your exponent be positive or negative? For numbers greater than 1, the exponent on 10 will be positive when you are using scientific notation. Refer to the table presented above:

2. You are writing a number that is between 0 and 1 in scientific notation. Will your exponent be positive or negative? We can reason that since numbers greater than 1 will have a positive exponent, numbers between 0 and 1 will have a negative exponent. Why are we specifying numbers between 0 and 1? The numbers between 0 and 1 represent amounts that are fractional. Recall that we defined numbers with a negative exponent as $10^{-2}$ we have $\frac{1}{10\times10}=\frac{1}{100}$ which is a number between 0 and 1.

3. What power do you need to put on 10 to get a result of 1? Recall that any number or variable with an exponent of 0 is equal to 1, as in this example:

$\begin{array}{c}\frac{t^{8}}{t^{8}}=\frac{\cancel{t^{8}}}{\cancel{t^{8}}}=1\\\frac ParseError: EOF expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/04:_Exponents/4.04:_Scientific_Notation), /content/body/div[2]/div[3]/div[2]/div/p[10]/span[1], line 1, column 2 ParseError: EOF expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/04:_Exponents/4.04:_Scientific_Notation), /content/body/div[2]/div[3]/div[2]/div/p[10]/span[2], line 1, column 2 ={t}^{8 - 8}={t}^{0}\\\text{ therefore }\\{t}^{0}=1\end{array}$

We now have described the notation necessary to write all possible numbers on the number line in scientific notation.

In the next video you will see how to convert a number written in scientific notation into decimal notation.

$Thumbnail for the embedded element "Examples: Writing a Number in Decimal Notation When Given in Scientific Notation"$

Multiplying and Dividing Numbers Expressed in Scientific Notation

Numbers that are written in scientific notation can be multiplied and divided rather simply by taking advantage of the properties of numbers and the rules of exponents that you may recall. To multiply numbers in scientific notation, first multiply the numbers that aren’t powers of 10 (the a in $a\times10^{n}$). Then multiply the powers of ten by adding the exponents.

This will produce a new number times a different power of 10. All you have to do is check to make sure this new value is in scientific notation. If it isn’t, you convert it.

Let’s look at some examples.

$\left(3\times10^{8}\right)\left(6.8\times10^{-13}\right)$

[reveal-answer q=”395606″]Show Solution[/reveal-answer] [hidden-answer a=”395606″]Regroup using the commutative and associative properties.

$\left(3\times6.8\right)\left(10^{8}\times10^{-13}\right)$

Multiply the coefficients.

$\left(20.4\right)\left(10^{8}\times10^{-13}\right)$

Multiply the powers of 10 using the Product Rule. Add the exponents.

$20.4\times10^{-5}$

Convert 20.4 into scientific notation by moving the decimal point one place to the left and multiplying by $10^{1}$.

$\left(2.04\times10^{1}\right)\times10^{-5}$

Group the powers of 10 using the associative property of multiplication.

$2.04\times\left(10^{1}\times10^{-5}\right)$

Multiply using the Product Rule—add the exponents.

$2.04\times10^{1+\left(-5\right)}$

$\left(3\times10^{8}\right)\left(6.8\times10^{-13}\right)=2.04\times10^{-4}$

$\left(8.2\times10^{6}\right)\left(1.5\times10^{-3}\right)\left(1.9\times10^{-7}\right)$

[reveal-answer q=”23947″]Show Solution[/reveal-answer] [hidden-answer a=”23947″]Regroup using the commutative and associative properties.

$\left(8.2\times1.5\times1.9\right)\left(10^{6}\times10^{-3}\times10^{-7}\right)$

Multiply the numbers.

$\left(23.37\right)\left(10^{6}\times10^{-3}\times10^{-7}\right)$

Multiply the powers of 10 using the Product Rule—add the exponents.

$23.37\times10^{-4}$

Convert 23.37 into scientific notation by moving the decimal point one place to the left and multiplying by $10^{1}$.

$\left(2.337\times10^{1}\right)\times10^{-4}$

$2.337\times\left(10^{1}\times10^{-4}\right)$

Multiply using the Product Rule and add the exponents.

$2.337\times10^{1+\left(-4\right)}$

$\left(8.2\times10^{6}\right)\left(1.5\times10^{-3}\right)\left(1.9\times10^{-7}\right)=2.337\times10^{-3}$

In the following video you will see an example of how to multiply tow numbers that are written in scientific notation.

$Thumbnail for the embedded element "Examples: Multiplying Numbers Written in Scientific Notation"$

In order to divide numbers in scientific notation, you once again apply the properties of numbers and the rules of exponents. You begin by dividing the numbers that aren’t powers of 10 (the a in $a\times10^{n}$. Then you divide the powers of ten by subtracting the exponents.

This will produce a new number times a different power of 10. If it isn’t already in scientific notation, you convert it, and then you’re done.

$\displaystyle \frac{2.829\times 1 ParseError: EOF expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/04:_Exponents/4.04:_Scientific_Notation), /content/body/div[3]/div[3]/p[1]/span[1], line 1, column 2 }{3.45\times 1 ParseError: EOF expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/04:_Exponents/4.04:_Scientific_Notation), /content/body/div[3]/div[3]/p[1]/span[2], line 1, column 2 }$

[reveal-answer q=”364796″]Show Solution[/reveal-answer] [hidden-answer a=”364796″]Regroup using the associative property.

$\displaystyle \left( \frac{2.829}{3.45} \right)\left( \frac ParseError: colon expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/04:_Exponents/4.04:_Scientific_Notation), /content/body/div[3]/div[3]/p[3]/span[1], line 1, column 4 ParseError: colon expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/04:_Exponents/4.04:_Scientific_Notation), /content/body/div[3]/div[3]/p[3]/span[2], line 1, column 4 \right)$

Divide the coefficients.

$\displaystyle \left(0.82\right)\left( \frac ParseError: colon expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/04:_Exponents/4.04:_Scientific_Notation), /content/body/div[3]/div[3]/p[5]/span[1], line 1, column 4 ParseError: colon expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/04:_Exponents/4.04:_Scientific_Notation), /content/body/div[3]/div[3]/p[5]/span[2], line 1, column 4 \right)$

Divide the powers of 10 using the Quotient Rule. Subtract the exponents.

$\begin{array}{l}0.82\times10^{-9-\left(-3\right)}\\0.82\times10^{-6}\end{array}$

Convert 0.82 into scientific notation by moving the decimal point one place to the right and multiplying by $10^{-1}$.

$\left(8.2\times10^{-1}\right)\times10^{-6}$

Group the powers of 10 together using the associative property.

$8.2\times\left(10^{-1}\times10^{-6}\right)$

$8.2\times10^{-1+\left(-6\right)}$

$\displaystyle \frac{2.829\times ParseError: EOF expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/04:_Exponents/4.04:_Scientific_Notation), /content/body/div[3]/div[3]/div/p[1]/span[1], line 1, column 3 }{3.45\times ParseError: EOF expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/04:_Exponents/4.04:_Scientific_Notation), /content/body/div[3]/div[3]/div/p[1]/span[2], line 1, column 3 }=8.2\times ParseError: EOF expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/04:_Exponents/4.04:_Scientific_Notation), /content/body/div[3]/div[3]/div/p[1]/span[3], line 1, column 3 $

$\displaystyle \frac{\left(1.37\times10^{4}\right)\left(9.85\times10^{6}\right)}{5.0\times10^{12}}$

[reveal-answer q=”337143″]Show Solution[/reveal-answer] [hidden-answer a=”337143″]Regroup the terms in the numerator according to the associative and commutative properties.

$\displaystyle \frac{\left( 1.37\times 9.85 \right)\left( ParseError: EOF expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/04:_Exponents/4.04:_Scientific_Notation), /content/body/div[3]/div[4]/p[3]/span[1], line 1, column 3 \times ParseError: EOF expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/04:_Exponents/4.04:_Scientific_Notation), /content/body/div[3]/div[4]/p[3]/span[2], line 1, column 3 \right)}{5.0\times ParseError: EOF expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/04:_Exponents/4.04:_Scientific_Notation), /content/body/div[3]/div[4]/p[3]/span[3], line 1, column 3 }$

$\displaystyle \frac{13.4945\times ParseError: EOF expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/04:_Exponents/4.04:_Scientific_Notation), /content/body/div[3]/div[4]/p[5]/span[1], line 1, column 3 }{5.0\times ParseError: EOF expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/04:_Exponents/4.04:_Scientific_Notation), /content/body/div[3]/div[4]/p[5]/span[2], line 1, column 3 }$

Regroup using the associative property.

$\displaystyle \left( \frac{13.4945}{5.0} \right)\left( \frac ParseError: colon expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/04:_Exponents/4.04:_Scientific_Notation), /content/body/div[3]/div[4]/p[7]/span[1], line 1, column 4 ParseError: colon expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/04:_Exponents/4.04:_Scientific_Notation), /content/body/div[3]/div[4]/p[7]/span[2], line 1, column 4 \right)$

Divide the numbers.

$\displaystyle \left(2.6989\right)\left(\frac{10^{10}}{10^{12}}\right)$

Divide the powers of 10 using the Quotient Rule—subtract the exponents.

$\displaystyle \begin{array}{c}\left(2.6989 \right)\left( ParseError: EOF expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/04:_Exponents/4.04:_Scientific_Notation), /content/body/div[3]/div[4]/p[11]/span[1], line 1, column 3 \right)\\2.6989\times ParseError: EOF expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/04:_Exponents/4.04:_Scientific_Notation), /content/body/div[3]/div[4]/p[11]/span[2], line 1, column 3 \end{array}$

$\displaystyle \frac{\left( 1.37\times ParseError: EOF expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/04:_Exponents/4.04:_Scientific_Notation), /content/body/div[3]/div[4]/div/p[1]/span[1], line 1, column 3 \right)\left( 9.85\times ParseError: EOF expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/04:_Exponents/4.04:_Scientific_Notation), /content/body/div[3]/div[4]/div/p[1]/span[2], line 1, column 3 \right)}{5.0\times ParseError: EOF expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/04:_Exponents/4.04:_Scientific_Notation), /content/body/div[3]/div[4]/div/p[1]/span[3], line 1, column 3 }=2.6989\times ParseError: EOF expected (click for details) Callstack: at (Courses/Lumen_Learning/Beginning_Algebra_(Lumen)/04:_Exponents/4.04:_Scientific_Notation), /content/body/div[3]/div[4]/div/p[1]/span[4], line 1, column 3 $

Notice that when you divide exponential terms, you subtract the exponent in the denominator from the exponent in the numerator. You will see another example of dividing numbers written in scientific notation in the following video.

$Thumbnail for the embedded element "Examples: Dividing Numbers Written in Scientific Notation"$

Solve application problems

$Molecule of water with one oxygen bonded to two hydrogen.$

Learning rules for exponents seems pointless without context, so let’s explore some examples of using scientific notation that involve real problems. First, let’s look at an example of how scientific notation can be used to describe real measurements.

Match each length in the table with the appropriate number of meters described in scientific notation below.

[reveal-answer q=”993302″]Show Solution[/reveal-answer] [hidden-answer a=”993302″]

$Red Blood Cells.$

One of the most important parts of solving a “real” problem is translating the words into appropriate mathematical terms, and recognizing when a well known formula may help. Here’s an example that requires you to find the density of a cell, given its mass and volume. Cells aren’t visible to the naked eye, so their measurements, as described with scientific notation, involve negative exponents.

Human cells come in a wide variety of shapes and sizes. The mass of an average human cell is about $10^{-6}\text{ meters }^3$. [3] Biologists have recently discovered how to use the density of some types of cells to indicate the presence of disorders such as sickle cell anemia or leukemia. [4] Density is calculated as the ratio of $\frac{\text{ mass }}{\text{ volume }}$. Calculate the density of an average human cell.

[reveal-answer q=”856454″]Show Solution[/reveal-answer] [hidden-answer a=”856454″]

Read and Understand: We are given an average cellular mass and volume as well as the formula for density. We are looking for the density of an average human cell.

Define and Translate: $v=\text{volume}=10^{-6}\text{ meters}^3$, $\text{density}=\frac{\text{ mass }}{\text{ volume }}$

Write and Solve: Use the quotient rule to simplify the ratio.

$\begin{array}{c}\text{ density }=\frac{2\times10^{-11}\text{ grams }}{10^{-6}\text{ meters }^3}\\\text{ }\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=2\times10^{-11-\left(-6\right)}\frac{\text{ grams }}{\text{ meters }^3}\\\text{ }\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=2\times10^{-5}\frac{\text{ grams }}{\text{ meters }^3}\\\end{array}$

If scientists know the density of healthy cells, they can compare the density of a sick person’s cells to that to rule out or test for disorders or diseases that may affect cellular density.

The average density of a human cell is $2\times10^{-5}\frac{\text{ grams }}{\text{ meters }^3}$

The following video provides an example of how to find the number of operations a computer can perform in a very short amount of time.

$Thumbnail for the embedded element "Application of Scientific Notation - Quotient 2 (Time for Computer Operations)"$

In the next example, you will use another well known formula, $d=r\cdot{t}$, to find how long it takes light to travel from the sun to the earth. Unlike the previous example, the distance between the earth and the sun is massive, so the numbers you will work with have positive exponents.

The speed of light is $1.5\times10^{11}$ meters from earth, how many seconds does it take for sunlight to reach the earth? Write your answer in scientific notation. [reveal-answer q=”532092″]Show Solution[/reveal-answer] [hidden-answer a=”532092″]

Read and Understand: We are looking for how long—an amount of time. We are given a rate which has units of meters per second and a distance in meters. This is a $d=r\cdot{t}$ problem.

Define and Translate:

$\begin{array}{l}d=1.5\times10^{11}\\r=3\times10^{8}\frac{\text{ meters }}{\text{ second }}\\t=\text{ ? }\end{array}$

Write and Solve: Substitute the values we are given into the $d=r\cdot{t}$ equation. We will work without units to make it easier. Often, scientists will work with units to make sure they have made correct calculations.

$\begin{array}{c}d=r\cdot{t}\\1.5\times10^{11}=3\times10^{8}\cdot{t}\end{array}$

Divide both sides of the equation by $3\times10^{8}$ to isolate t.

$\begin{array}{c}1.5\times10^{11}=3\times10^{8}\cdot{t}\\\text{ }\\\frac{1.5\times10^{11}}{3\times10^{8}}=\frac{3\times10^{8}}{3\times10^{8}}\cdot{t}\end{array}$

On the left side, you will need to use the quotient rule of exponents to simplify, and on the right, you are left with t.

$\begin{array}{c}\frac{1.5\times10^{11}}{3\times10^{8}}=\frac{3\times10^{8}}{3\times10^{8}}\cdot{t}\\\text{ }\\\left(\frac{1.5}{3}\right)\times\left(\frac{10^{11}}{10^{8}}\right)=t\\\text{ }\\\left(0.5\right)\times\left(10^{11-8}\right)=t\\0.5\times10^3=t\end{array}$

This answer is not in scientific notation, so we will move the decimal to the right, which means we need to subtract one factor of 10.

$0.5\times10^3=5.0\times10^2=t$

The time it takes light to travel from the sun to the earth is $5.0\times10^2=t$ seconds, or in standard notation, 500 seconds. That’s not bad considering how far it has to travel!

In the following video we calculate how many miles the participants of the New York marathon ran combined, and compare that to the circumference of the earth.

$Thumbnail for the embedded element "Application of Scientific Notation - Quotient 1 (Number of Times Around the Earth)"$

Large and small numbers can be written in scientific notation to make them easier to understand. In the next section, you will see that performing mathematical operations such as multiplication and division on large and small numbers is made easier by scientific notation and the rules of exponents.

Scientific notation was developed to assist mathematicians, scientists, and others when expressing and working with very large and very small numbers. Scientific notation follows a very specific format in which a number is expressed as the product of a number greater than or equal to one and less than ten, and a power of 10. The format is written $1\leq{a}<10$ and n is an integer. To multiply or divide numbers in scientific notation, you can use the commutative and associative properties to group the exponential terms together and apply the rules of exponents.

Orders of magnitude (mass). (n.d.). Retrieved May 26, 2016, from https://en.Wikipedia.org/wiki/Orders_of_magnitude_(mass) ↵
How Big is a Human Cell? ↵
How big is a human cell? - Weizmann Institute of Science. (n.d.). Retrieved May 26, 2016, from www.weizmann.ac.il/plants/Milo/images/humanCellSize120116Clean.pdf↵
Grover, W. H., Bryan, A. K., Diez-Silva, M., Suresh, S., Higgins, J. M., & Manalis, S. R. (2011). Measuring single-cell density. Proceedings of the National Academy of Sciences, 108(27), 10992-10996. doi:10.1073/pnas.1104651108 ↵
Revision and Adaptation. Provided by : Lumen Learning. License : CC BY: Attribution
Application of Scientific Notation - Quotient 1 (Number of Times Around the Earth). Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/15tw4-v100Y . License : CC BY: Attribution
Application of Scientific Notation - Quotient 2 (Time for Computer Operations). Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/Cbm6ejEbu-o . License : CC BY: Attribution
Screenshot: water molecule. Provided by : Lumen Learning. License : CC BY: Attribution
Screenshot: red blood cells. Provided by : Lumen Learning. License : CC BY: Attribution
Screenshot: light traveling from the sun to the earth. Provided by : Lumen Learning. License : CC BY: Attribution
Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program. Provided by : Monterey Institute of Technology and Education. Located at : nrocnetwork.org/resources/downloads/nroc-math-open-textbook-units-1-12-pdf-and-word-formats/. License : CC BY: Attribution
Examples: Write a Number in Scientific Notation. Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/fsNu3AdIgdk . License : CC BY: Attribution
Examples: Writing a Number in Decimal Notation When Given in Scientific Notation. Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/8BX0oKUMIjw . License : CC BY: Attribution
Examples: Dividing Numbers Written in Scientific Notation. Authored by : James Sousa (Mathispower4u.com) for Lumen Learning. Located at : https://youtu.be/RlZck2W5pO4 . License : CC BY: Attribution
Examples: Multiplying Numbers Written in Scientific Notation. Authored by : James Sousa (Mathispower4u.com) . Located at : https://youtu.be/5ZAY4OCkp7U . License : CC BY: Attribution

Scientific Notation Questions with Solutions

Questions on scientific notation are presented along with answers and detailed solutions . Scientific notation of a number is of form: $ a \times 10^n $ where $ 1 \le a \lt 10 $, $ n $ is an integer and $ a $ is called the mantissa.

$ 5.9 \times 10^{-6} - 2.0 \times 10^{-7} = $ A) $ \quad 5.7 \times 10^{-7} $ B) $ \quad 5.7 \times 10^{-6} $ C) $ \quad 3.9 \times 10^{-6} $ D) $ \quad 7.9 \times 10^{-6} $ E) $ \quad 5.7 \times 10^{-5} $

Solutions to the Above Questions

Answer E Solution In $ a $, the mantissa $ 23.7 $ is greater than $10 $, therefore it is not in scientific notation. In $ c $, the base $ 5 $ must be base $ 10 $. In $ d $, the mantissa $ 0.3 $ is less than $1 $, therefore it is not in scientific notation. In $ e $, the mantissa $ 10.0 $ is equal to $10 $, therefore it is not in scientific notation.
Answer E Solution Scientific notation of a number is of form: $ a \times 10^n $ where $ 1 \le a \lt 10 $ and $ n $ is an integer. $ 510000 $ is bigger than $ 10 $, therefore start with a decimal point from the right $ 510000. $ and move it till the number is between $ 1 $ and $ 10 $ not included. Hence for 510000, if we start with a decimal point from the right, we need to move the decimal point n = 5 times in order to obtain $ 5.10000 $ which is a number between $ 1 $ and $ 10 $ not included. $ 510,000 $ is written in scientific notation as: $ 5.1 \times 10^{5} $
Answer A Solution We first add the numbers: $ 100000 + 3000000 = 3100000 $ $ 3100000 $ is bigger than 10, we therefore start with a decimal point on the right and move it $ n = 6 $ times to obtain $ 3.1 $ which is a number between $ 1 $ and $ 10 $ not included. Hence $ 100000 + 3,000000 = 3.1 \times 10^{6} $
Answer C Solution The given number $ 0.0000028 $ is smaller than one and in order to obtain a number between $ 1 $ and $ 10 $ not included, we need to move the decimal point to the RIGHT. We need to move the decimal point $ n = 6 $ times in order to write the given number as $ 2.8 $, between $ 1 $ and $ 10 $ not included Hence $ 0.0000028 $ is written in scientific notation as: $ 2.8 \times 10^{-6} $
Answer D Solution In this question, we are given the scientific notation of a number and asked to write it in standard form. Hence $ 1.2 \times 10^6 = 1.2 \times 1000000 = 1200000 $
Answer B Solution We are given the scientific notation: $ 2.3 \times 10^{-5} = \dfrac{2.3}{100000} = 0.000023 $
Answer E Solution Write in decimal form: $ \quad \dfrac{1}{10000}+\dfrac{2}{1000000} = 0.0001 + 0.000002 = 0.000102 $ In scientific notation: $ \quad \dfrac{1}{10000}+\dfrac{2}{1000000} = 0.000102 = 1.02 \times 10^{-4} $
Answer A Solution Write in decimal form: $ \quad 2.1 \times 10^{-5} + 3.0 \times 10^{-4} = 0.000021 + 0.0003 = 0.000321 $ In scientific notation: $ \quad 2.1 \times 10^{-5} + 3.0 \times 10^{-4} = 0.000021 + 0.0003 = 0.000321 = 3.21 \times 10^{-4} $
Answer C Solution The mantissan $ 120.054 $ is larger than $ 1 $, hence we need to move the decimal point $ n = 2 $ to the left to rewrite $ 120.054 $ in decimal form as $ 1.20054 \times 10^2 $ Hence $ \quad 120.054 \times 10^{-6} = 1.20054 \times 10^2 \times 10^{-6} = 1.20054 \times 10^{-4} $
Answer B Solution Write in decimal form: $ \quad 5.9 \times 10^{-6} - 2.0 \times 10^{-7} = 0.0000059 - 0.0000002 = 0.0000057 $ In scientific notation: $ \quad 0.0000057 = 5.7 \times 10^{-6} $

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HiSET: Math : Solve problems using scientific notation

Study concepts, example questions & explanations for hiset: math, all hiset: math resources, example questions, example question #1 : solve problems using scientific notation.

Simplify the following expression using scientific notation.