11.1.4: Scientific Notation

Last updated
Save as PDF

Page ID: 67618

The NROC Project

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$ \newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\id}{\mathrm{id}}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\kernel}{\mathrm{null}\,}$

$ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$

$ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$

$ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\AA}{\unicode[.8,0]{x212B}}$

$ \newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$ \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$ \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vectorC}[1]{\textbf{#1}} $

$ \newcommand{\vectorD}[1]{\overrightarrow{#1}} $

$ \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} $

$ \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} $

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$\newcommand{\avec}{\mathbf a}$ $\newcommand{\bvec}{\mathbf b}$ $\newcommand{\cvec}{\mathbf c}$ $\newcommand{\dvec}{\mathbf d}$ $\newcommand{\dtil}{\widetilde{\mathbf d}}$ $\newcommand{\evec}{\mathbf e}$ $\newcommand{\fvec}{\mathbf f}$ $\newcommand{\nvec}{\mathbf n}$ $\newcommand{\pvec}{\mathbf p}$ $\newcommand{\qvec}{\mathbf q}$ $\newcommand{\svec}{\mathbf s}$ $\newcommand{\tvec}{\mathbf t}$ $\newcommand{\uvec}{\mathbf u}$ $\newcommand{\vvec}{\mathbf v}$ $\newcommand{\wvec}{\mathbf w}$ $\newcommand{\xvec}{\mathbf x}$ $\newcommand{\yvec}{\mathbf y}$ $\newcommand{\zvec}{\mathbf z}$ $\newcommand{\rvec}{\mathbf r}$ $\newcommand{\mvec}{\mathbf m}$ $\newcommand{\zerovec}{\mathbf 0}$ $\newcommand{\onevec}{\mathbf 1}$ $\newcommand{\real}{\mathbb R}$ $\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$ $\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$ $\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$ $\newcommand{\laspan}[1]{\text{Span}\{#1\}}$ $\newcommand{\bcal}{\cal B}$ $\newcommand{\ccal}{\cal C}$ $\newcommand{\scal}{\cal S}$ $\newcommand{\wcal}{\cal W}$ $\newcommand{\ecal}{\cal E}$ $\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$ $\newcommand{\gray}[1]{\color{gray}{#1}}$ $\newcommand{\lgray}[1]{\color{lightgray}{#1}}$ $\newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\row}{\text{Row}}$ $\newcommand{\col}{\text{Col}}$ $\renewcommand{\row}{\text{Row}}$ $\newcommand{\nul}{\text{Nul}}$ $\newcommand{\var}{\text{Var}}$ $\newcommand{\corr}{\text{corr}}$ $\newcommand{\len}[1]{\left|#1\right|}$ $\newcommand{\bbar}{\overline{\bvec}}$ $\newcommand{\bhat}{\widehat{\bvec}}$ $\newcommand{\bperp}{\bvec^\perp}$ $\newcommand{\xhat}{\widehat{\xvec}}$ $\newcommand{\vhat}{\widehat{\vvec}}$ $\newcommand{\uhat}{\widehat{\uvec}}$ $\newcommand{\what}{\widehat{\wvec}}$ $\newcommand{\Sighat}{\widehat{\Sigma}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\newcommand{\amp}{&}$ $\definecolor{fillinmathshade}{gray}{0.9}$

Learning Objectives

Convert between scientific and decimal notation.
Multiply and divide using scientific notation.
Solve application problems.

Introduction

When working with very large or very small numbers, scientists, mathematicians, and engineers often use scientific notation to express those quantities. Scientific notation uses exponential notation. The following are examples of scientific notation.

Light year: number of miles light travels in one year, about 5,880,000,000,000

Scientific notation is $\ 5.88 \times 10^{12}$ miles.

hydrogen atom: has a diameter of about 0.00000005 millimeters

Scientific notation is $\ 5 \times 10^{-8}$ millimeters

Computation with very large numbers is made easier with scientific notation.

Learning to Use Scientific Notation

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.

Since it’s so useful, let’s look more closely at the details of scientific notation format.

Scientific Notation

A positive number is written in scientific notation if it is written as $\ a \times 10^{n}$ where the coefficient $\ a$ has a value such that $\ 1 \leq a<10$ and $\ n$ is an integer.

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

Number	Scientific Notation?	Explanation
$\ 1.85 \times 10^{-2}$	yes	$\ 1 \leq 1.85<10$ -2 is an integer
$\ 1.083 \times 10^{\frac{1}{2}}$	no	$\ \frac{1}{2}$ is not an integer
$\ 0.82 \times 10^{14}$	no	$\ 0.82$ is not $\ \geq 1$
$\ 10 \times 10^{3}$	no	$\ 10$ is not $\ <10$

Exercise

Which number below is written in scientific notation?

$\ 4.25 \times 10^{0.08}$
$\ 0.425 \times 10^{7}$
$\ 42.5 \times 10^{5}$
$\ 4.25 \times 10^{6}$

Answer

Incorrect. The exponent must be an integer and 0.08 is not an integer. The correct answer is $\ 4.25 \times 10^{6}$.
Incorrect. This is not in scientific notation because 0.425 is less than 1. The correct answer is $\ 4.25 \times 10^{6}$.
Incorrect. This is not in scientific notation because 42.5 is greater than 10. The correct answer is $\ 4.25 \times 10^{6}$.
Correct. This is scientific notation. 4.25 is greater than 1 and less than 10, and 6 is an integer.

Writing Decimal Notation 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.

Large Numbers
Decimal Notation	Scientific Notation
500.0	$\ 5 \times 10^{2}$
80,000.0	$\ 8 \times 10^{4}$
$\ 43,000,000.0$	$\ 4.3 \times 10^{7}$
$\ 62,500,000,000.0$	$\ 6.25 \times 10^{10}$

Small Numbers
Decimal Notation	Scientific Notation
$\ 0.05$	$\ 5 \times 10^{-2}$
$\ 0.0008$	$\ 8 \times 10^{-4}$
$\ 0.00000043$	$\ 4.3 \times 10^{-7}$
$\ 0.000000000625$	$\ 6.25 \times 10^{-10}$

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.

$\ \begin{array}{r}
180,000{\color{red}.}=18,000{\color{red}.}0 \times 10^{1} \ \ \ \ \ \ \ \ \\
1,800{\color{red}.}00 \times 10^{2} \ \ \ \ \ \ \\
180{\color{red}.}000 \times 10^{3} \ \ \ \ \\
18{\color{red}.}0000 \times 10^{4} \ \ \\
1{\color{red}.}80000 \times 10^{5}
\end{array}$

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

Exercise

The world population is estimated to be about 6,800,000,000 people. Which answer expresses this number in scientific notation?

$\ 7 \times 10^{9}$
$\ 0.68 \times 10^{10}$
$\ 6.8 \times 10^{9}$
$\ 68 \times 10^{8}$

Answer

Incorrect. Scientific notation rewrites numbers, it doesn’t round them. The correct answer is $\ 6.8 \times 10^{9}$.
Incorrect. Although $\ 0.68 \times 10^{10}$ is equivalent to 6,800,000,000, 0.68 is not the form for scientific notation as 0.68 is not a number between 1 and 10. The correct answer is $\ 6.8 \times 10^{9}$.
Correct. The number $\ 6.8 \times 10^{9}$ is equivalent to 6,800,000,000 and uses the proper format for each factor.
Incorrect. Although $\ 68 \times 10^{8}$ is equivalent to 6,800,000,000, it is not written in scientific notation as 68 is not between 1 and 10. The correct answer is $\ 6.8 \times 10^{9}$.

Advanced Question

Represent $\ 1.00357 \times 10^{-6}$ in decimal form.

1.00357000000
0.000100357
0.000001357
0.00000100357

Answer

Incorrect. You added six 0s to the end of the decimal. This does not change its value at all; 1.00357000000=1.00357, which is not equal to $\ 1.00357 \times 10^{-6}$. The correct answer is 0.00000100357.
Incorrect. You moved the decimal point in the correct direction, but you did not move it enough places. The correct answer is 0.00000100357.
Incorrect. You moved the decimal point the correct number of spaces, but the number you created is different than the number you started with: $\ 1.00357 \times 10^{-6} \neq 0.000001357$. Remember that the zeros in between 1 and 3 must also be included in the final number. The correct answer is 0.00000100357.
Correct. The exponent is -6. You moved the decimal point 6 spots to the left, creating the decimal 0.00000100357.

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.

$\ \begin{array}{r}
0{\color{red}.}00004=00{\color{red}.}0004 \times 10^{-1} \\
000{\color{red}.}004 \times 10^{-2}\ \ \\
0000{\color{red}.}04 \times 10^{-3} \ \ \ \ \\
00000{\color{red}.}4 \times 10^{-4} \ \ \ \ \ \ \\
000004 {\color{red}.} \times 10^{-5}\ \ \ \ \ \ \ \
\end{array}$

$\ 0.00004=4 \times 10^{-5}$

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

Writing Scientific Notation in Decimal Notation

You can also write scientific notation as decimal notation. For example, the number of miles that light travels in a year is $\ 5.88 \times 10^{12}$. $\ 5.88 \times 10^{12}$, and a hydrogen atom has a diameter of $\ 5 \times 10^{-8}$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.

$Screen Shot 2021-06-08 at 11.33.00 PM.png$

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.

Exercise

Rewrite $\ 1.57 \times 10^{-10}$ in decimal notation.

15,700,000,000
0.000000000157
0.0000000000157
$\ 157 \times 10^{-12}$

Answer

Incorrect. You moved the decimal point in the wrong direction. The exponent is negative, so to convert to decimal format, move the decimal point to the left, not the right. The correct answer is 0.000000000157.
Correct. The expression has a negative exponent, so you move the decimal point 10 places to the left to convert it to decimal notation, 0.000000000157.
Incorrect. You inserted 10 zeros between the number and the decimal point. Move the decimal point 10 places to the left instead. The correct answer is 0.000000000157.
Incorrect. This number is equivalent to the original number, but it is not in decimal notation. The correct answer is 0.000000000157.

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 $\ a \times 10^{n}$). 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.

Example

$\ \left(3 \times 10^{8}\right)\left(6.8 \times 10^{-13}\right)$.

Solution

$\ (3 \times 6.8)\left(10^{8} \times 10^{-13}\right)$	Regroup, using the commutative and associative properties.
$\ (20.4)\left(10^{8} \times 10^{-13}\right)$	Multiply the coefficients.
$\ 20.4 \times 10^{-5}$	Multiply the powers of 10 using the Product Rule: add the exponents.
$\ \left(2.04 \times 10^{1}\right) \times 10^{-5}$	Convert 20.4 into scientific notation by moving the decimal point one place to the left and multiplying by 10¹.
$\ 2.04 \times\left(10^{1} \times 10^{-5}\right)$	Group the powers of 10 using the associative property of multiplication.
$\ 2.04 \times 10^{1+(-5)}$	Multiply using the Product Rule: add the exponents.

$\ \left(3 \times 10^{8}\right)\left(6.8 \times 10^{-13}\right)=2.04 \times 10^{-4}$

Advanced Example

$\ \left(8.2 \times 10^{6}\right)\left(1.5 \times 10^{-3}\right)\left(1.9 \times 10^{-7}\right)$

Solution

$\ (8.2 \times1.5\times1.9)(10^6\times10^{-3}\times10^{-7})$	Regroup, using the commutative and associative properties.
$\ (23.37)\left(10^{6} \times 10^{-3} \times 10^{-7}\right)$	Multiply the numbers.
$\ 23.37 \times 10^{-4}$	Multiply the powers of 10 using the Product Rule: add the exponents.
$\ \left(2.337 \times 10^{1}\right) \times 10^{-4}$	Convert 23.37 into scientific notation by moving the decimal point one place to the left and multiplying by 10¹.
$\ 2.337 \times\left(10^{1} \times 10^{-4}\right)$	Group the powers of 10 using the associative property of multiplication.
$\ 2.337 \times 10^{1+(-4)}$	Multiply using the Product Rule and add the exponents.

$\ \left(8.2 \times 10^{6}\right)\left(1.5 \times 10^{-3}\right)\left(1.9 \times 10^{-7}\right)=2.337 \times 10^{-3}$

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 $\ a \times 10^{n}$). Then you divide the powers of 10 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.

Let’s look at some examples.

Example

$\ \frac{2.829 \times 10^{-9}}{3.45 \times 10^{-3}}$

Solution

$\ \left(\frac{2.829}{3.45}\right)\left(\frac{10^{-9}}{10^{-3}}\right)$	Regroup, using the associative property.
$\ (0.82)\left(\frac{10^{-9}}{10^{-3}}\right)$	Divide the coefficients.
$\ \begin{array}{c} 0.82 \times 10^{-9-(-3)} \\ 0.82 \times 10^{-6} \end{array}$	Divide the powers of 10 using the Quotient Rule: subtract the exponents.
$\ \left(8.2 \times 10^{-1}\right) \times 10^{-6}$	Convert 0.82 into scientific notation by moving the decimal point one place to the right and multiplying by 10^-1.
$\ 8.2 \times\left(10^{-1} \times 10^{-6}\right)$	Group the powers of 10 together using the associative property.
$\ 8.2 \times 10^{-1+(-6)}$	Multiply the powers of 10 using the Product Rule: add the exponents.

$\ \frac{2.829 \times 10^{-9}}{3.45 \times 10^{-3}}=8.2 \times 10^{-7}$

Advanced Example

$\ \frac{\left(1.37 \times 10^{4}\right)\left(9.85 \times 10^{6}\right)}{5.0 \times 10^{12}}$

Solution

$\ \frac{(1.37 \times 9.85)\left(10^{6} \times 10^{4}\right)}{5.0 \times 10^{12}}$	Regroup the terms in the numerator according to the associative and commutative properties.
$\ \frac{13.4945 \times 10^{10}}{5.0 \times 10^{12}}$	Multiply.
$\ \left(\frac{13.4945}{5.0}\right)\left(\frac{10^{10}}{10^{12}}\right)$	Regroup, using the associative property.
$\ (2.6989)\left(\frac{10^{10}}{10^{12}}\right)$	Divide the numbers.
$\ \begin{array}{r} (2.6989)\left(10^{10-12}\right) \\ 2.6989 \times 10^{-2} \end{array}$	Divide the powers of 10 using the Quotient Rule: subtract the exponents.

$\ \frac{\left(1.37 \times 10^{4}\right)\left(9.85 \times 10^{6}\right)}{5.0 \times 10^{12}}=2.6989 \times 10^{-2}$

Notice that when you divide exponential terms, you subtract the exponent in the denominator from the exponent in the numerator.

Exercise

Evaluate $\ \left(4 \times 10^{-10}\right)\left(3 \times 10^{5}\right)$ and express the result in scientific notation.

$\ 1.2 \times 10^{-4}$
$\ 12 \times 10^{-5}$
$\ 7 \times 10^{-5}$
$\ 1.2 \times 10^{-50}$

Answer

Correct. $\ 1.2 \times 10^{-4}$ is an accurate computation and correct scientific notation.
Incorrect. Almost correct, but now you have to convert the coefficient 12 into scientific notation. 12 is greater than 10 and scientific notation requires this number to be greater than or equal to 1 and less than 10. The correct answer is $\ 1.2 \times 10^{-4}$.
Incorrect. Multiply, not add, the numbers 4 and 3. The correct answer is $\ 1.2 \times 10^{-4}$.
Incorrect. Add, not multiply, exponents. The correct answer is $\ 1.2 \times 10^{-4}$.

Advanced Question

Evaluate $\ \left(3.15 \times 10^{4}\right)\left(5.15 \times 10^{-7}\right)$ and express the result as a decimal.

0.0162225
162225
0.000162225
16.2225

Answer

Correct. $\ 3.15 \times 5.15=16.2225$, and $\ 10^{4} \times 10^{-7}=10^{-3}$. The exponent is negative, so to convert to decimal format, move the decimal point three spaces to the left for a value of 0.0162225.
Incorrect. The expression has a negative exponent, so you move the decimal point 3 places to the left, not to the right, to convert it to decimal notation. The correct answer is 0.0162225.
Incorrect. You moved the decimal point in the correct direction, but not the correct number of places. Remember that $\ 10^{4} \times 10^{-7}=10^{-3}$, so you have to move the decimal point three places to the left. The correct answer is 0.0162225.
Incorrect. $\ 3.15 \times 5.15=16.2225$, but it looks like you forgot to multiply $\ 10^{4} \times 10^{-7}$. The correct answer is 0.0162225.

Summary

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 1 and less than 10, and a power of 10. The format is written $\ a \times 10^{n}$, where $\ 1 \leq a<10$ 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.

Number	Scientific Notation?	Explanation
\(\ 1.85 \times 10^{-2}\)	yes	\(\ 1 \leq 1.85<10\) -2 is an integer
\(\ 1.083 \times 10^{\frac{1}{2}}\)	no	\(\ \frac{1}{2}\) is not an integer
\(\ 0.82 \times 10^{14}\)	no	\(\ 0.82\) is not \(\ \geq 1\)
\(\ 10 \times 10^{3}\)	no	\(\ 10\) is not \(\ <10\)

Decimal Notation	Scientific Notation
500.0	\(\ 5 \times 10^{2}\)
80,000.0	\(\ 8 \times 10^{4}\)
\(\ 43,000,000.0\)	\(\ 4.3 \times 10^{7}\)
\(\ 62,500,000,000.0\)	\(\ 6.25 \times 10^{10}\)

Decimal Notation	Scientific Notation
\(\ 0.05\)	\(\ 5 \times 10^{-2}\)
\(\ 0.0008\)	\(\ 8 \times 10^{-4}\)
\(\ 0.00000043\)	\(\ 4.3 \times 10^{-7}\)
\(\ 0.000000000625\)	\(\ 6.25 \times 10^{-10}\)

Search

Text Color

Text Size

Margin Size

Font Type

Scientific Notation

Exercise

Exercise

Advanced Question

Exercise

Example

Advanced Example

Example

Advanced Example

Exercise

Advanced Question

\(\ (3 \times 6.8)\left(10^{8} \times 10^{-13}\right)\)	Regroup, using the commutative and associative properties.
\(\ (20.4)\left(10^{8} \times 10^{-13}\right)\)	Multiply the coefficients.
\(\ 20.4 \times 10^{-5}\)	Multiply the powers of 10 using the Product Rule: add the exponents.
\(\ \left(2.04 \times 10^{1}\right) \times 10^{-5}\)	Convert 20.4 into scientific notation by moving the decimal point one place to the left and multiplying by 10¹.
\(\ 2.04 \times\left(10^{1} \times 10^{-5}\right)\)	Group the powers of 10 using the associative property of multiplication.
\(\ 2.04 \times 10^{1+(-5)}\)	Multiply using the Product Rule: add the exponents.

\(\ (8.2 \times1.5\times1.9)(10^6\times10^{-3}\times10^{-7})\)	Regroup, using the commutative and associative properties.
\(\ (23.37)\left(10^{6} \times 10^{-3} \times 10^{-7}\right)\)	Multiply the numbers.
\(\ 23.37 \times 10^{-4}\)	Multiply the powers of 10 using the Product Rule: add the exponents.
\(\ \left(2.337 \times 10^{1}\right) \times 10^{-4}\)	Convert 23.37 into scientific notation by moving the decimal point one place to the left and multiplying by 10¹.
\(\ 2.337 \times\left(10^{1} \times 10^{-4}\right)\)	Group the powers of 10 using the associative property of multiplication.
\(\ 2.337 \times 10^{1+(-4)}\)	Multiply using the Product Rule and add the exponents.

\(\ \left(\frac{2.829}{3.45}\right)\left(\frac{10^{-9}}{10^{-3}}\right)\)	Regroup, using the associative property.
\(\ (0.82)\left(\frac{10^{-9}}{10^{-3}}\right)\)	Divide the coefficients.
\(\ \begin{array}{c} 0.82 \times 10^{-9-(-3)} \\ 0.82 \times 10^{-6} \end{array}\)	Divide the powers of 10 using the Quotient Rule: subtract the exponents.
\(\ \left(8.2 \times 10^{-1}\right) \times 10^{-6}\)	Convert 0.82 into scientific notation by moving the decimal point one place to the right and multiplying by 10^-1.
\(\ 8.2 \times\left(10^{-1} \times 10^{-6}\right)\)	Group the powers of 10 together using the associative property.
\(\ 8.2 \times 10^{-1+(-6)}\)	Multiply the powers of 10 using the Product Rule: add the exponents.

\(\ \frac{(1.37 \times 9.85)\left(10^{6} \times 10^{4}\right)}{5.0 \times 10^{12}}\)	Regroup the terms in the numerator according to the associative and commutative properties.
\(\ \frac{13.4945 \times 10^{10}}{5.0 \times 10^{12}}\)	Multiply.
\(\ \left(\frac{13.4945}{5.0}\right)\left(\frac{10^{10}}{10^{12}}\right)\)	Regroup, using the associative property.
\(\ (2.6989)\left(\frac{10^{10}}{10^{12}}\right)\)	Divide the numbers.
\(\ \begin{array}{r} (2.6989)\left(10^{10-12}\right) \\ 2.6989 \times 10^{-2} \end{array}\)	Divide the powers of 10 using the Quotient Rule: subtract the exponents.