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How to multiply scientific notation

# How to multiply scientific notation

Here you will learn about how to multiply with scientific notation including what it is and how to solve problems.

Students will first learn how to multiply with scientific notation as part of expressions and equations in 8 th grade.

## What is how to multiply scientific notation?

Multiplying with scientific notation is completing multiplication between two numbers that are written in scientific notation.

Scientific notation is writing numbers in this form:

a\times10^{n}

Where a is a number 1\leq{a}<10 and n is an integer (whole number).

Numbers written in scientific notation can make some calculations with very large numbers or small numbers neater and quicker to compute.

For example,

Solve 2.4\times{10^{5}}\times{5}\times{10^{-3}}.

Because multiplication is commutative (the order does not matter), reorder this expression to be 2.4\times{5}\times{10^{5}}\times{10^{-3}}.

(2.4\times{5})\times\left({10^5}\times{10^{-3}}\right)

Hint: Use the rule a^{b}\times{a^{c}}=a^{b+c} to simply the powers of 10.

\begin{aligned}&(2.4\times{5})\times\left({10^5}\times{10^{-3}}\right) \\\\ &=12\times{10^2} \end{aligned}

Since 12 is NOT between 1 and 10, adjust the power of 10.

\begin{aligned}& 12 \times 10^2 \\\\ & =(1.2 \times 10) \times 10^2 \\\\ & =1.2 \times\left(10 \times 10^2\right) \\\\ & =1.2 \times 10^3 \end{aligned}

## Common Core State Standards

How does this relate to 8 th grade math?

• Grade 8 – Expressions and Equations (8.EE.A.3)
Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.

For example, estimate the population of the United States as 3\times{10^8} and the population of the world as 7\times{10^9}, and determine that the world population is more than 20 times larger.

• Grade 8 – Expressions and Equations (8.EE.A.4)
Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used.

Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (For example, use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

## How to multiply with scientific notation

In order to multiply with scientific notation:

1. Multiply the non-zero numbers.
2. Multiply the powers of \bf{10} by adding or subtracting the exponents.
3. Write the solution in scientific notation.

## How to multiply scientific notation examples

### Example 1: non-zero whole numbers

Calculate \left(3\times{10^4}\right)\times\left(8\times{10^8}\right).

1. Multiply or divide the non-zero numbers.

3\times{8}=24

2Multiply the powers of \bf{10} by adding or subtracting the exponents.

{10^4}\times{10^8}=10^{4+8}=10^{12}

3Write the solution in scientific notation.

24\times 10^{12}

24 is not between 1 and 10.

Convert 24\times 10^{12} back to scientific notation.

\begin{aligned}& 24 \times 10^{12} \\\\ & =(2.4 \times 10) \times 10^{12} \\\\ & =2.4 \times 10^{13} \end{aligned}

\left(3 \times 10^4\right) \times\left(8 \times 10^8\right)=2.4 \times 10^{13}

### Example 2: non-zero numbers are rational

Calculate \left(2.8 \times 10^7\right) \times\left(5.4 \times 10^2\right).

Multiply or divide the non-zero numbers.

Multiply the powers of \bf{10} by adding or subtracting the exponents.

Write the solution in scientific notation.

### Example 3: multiplying by a decimal

Calculate \left(8.1 \times 10^5\right) \times\left(7.99 \times 10^{-2}\right).

Multiply or divide the non-zero numbers.

Multiply the powers of \bf{10} by adding or subtracting the exponents.

Write the solution in scientific notation.

### Example 4: a multiplicand with a negative exponent

Calculate \left(1.18 \times 10^{-7}\right) \times\left(4 \times 10^{15}\right).

Multiply or divide the non-zero numbers.

Multiply the powers of \bf{10} by adding or subtracting the exponents.

Write the solution in scientific notation.

### Example 5: word problem

The distance from Earth to Mars is 1.4 \times 10^6 miles. The closest star to Earth, Proxima Centauri, is 1.79 \times 10^6 times farther away. How far away is Proxima Centauri from Earth? Write your answer in scientific notation.

Multiply the non-zero numbers.

Multiply the powers of \bf{10} by adding or subtracting the exponents.

Write the solution in scientific notation.

### Example 6: word problem

A 12 inch piece of hair weighs around 2.3 \times 10^{-4} ounces. About how much would \text { 1. } 25 \times 10^5 hairs weigh? Write your answer in scientific notation.

Multiply the non-zero numbers.

Multiply the powers of \bf{10} by adding or subtracting the exponents.

Write the solution in scientific notation.

### Teaching tips for how to multiply scientific notation

• Review previous topics like exponential notation and exponent rules before teaching students how to multiply numbers with scientific notation.

• When choosing worksheets to use, look for ones that start with whole number coefficients before moving on to calculations with decimal numbers.

• To help students struggling with multiplying decimals, give them access to a multiplying decimals tutorial or let them use a scientific calculator when solving.

### Easy mistakes to make

• Not converting the first number, when it is too large or too small
After multiplying with scientific notation, check to see that the first part is 1\leq{n}<10. If it is greater than 10 or less than 1, use the powers of 10 to convert it.

• Not using the rules for powers of ten correctly when converting numbers
Since each place value is 10 times larger than the place to the right and 10 times smaller than the place to the left, this causes the decimal point to “move” a certain number of places as you divide or multiply the original number by multiples of ten.

For example,
In the number 0.79 \times 10^{-13}, \, 0.79 is not between 1 and 10. It needs to be converted.

\left(7.9 \times 10^{-1}\right) \times 10^{-13} \quad *Multiplying by 10^{-1} shifts the decimal places

\begin{aligned} & =7.9 \times\left(10^{-1} \times 10^{-13}\right) \\\\ & =7.9 \times\left(10^{-1+(-13)}\right) \\\\ & =7.9 \times 10^{-14} \end{aligned}

• Confusing the meaning of negative exponents
The positions to the right of the decimal are shown by powers of 10 with negative exponents, and each has an equivalent fraction.
For example,

\begin{aligned} 10^{-1}&=\cfrac{1}{10} \\\\ 10^{-2}&=\cfrac{1}{100} \\\\ 10^{-3}&=\cfrac{1}{1,000} \\\\ 10^{-4}&=\cfrac{1}{10,000} \end{aligned}

### Practice how to multiply scientific notation questions

1. Solve \left(3 \times 10^7\right) \times\left(7 \times 10^5\right). Write your answer in scientific notation.

2.1\times{10^{13}}

4\times{10^{12}}

21 \times 10^{35}

4.3 \times 10^2

Use the commutative property to rearrange the expression.

\left(3 \times 10^7\right) \times\left(7 \times 10^5\right)=(3 \times 7) \times\left(10^7 \times 10^5\right)

Now solve.

\begin{aligned}& (3 \times 7) \times\left(10^7 \times 10^5\right) \\\\ & =21 \times 10^{12}\end{aligned}

However, 21 \times 10^{12} is not in scientific notation since 21 is not between 1 and 10.

\begin{aligned}& 21 \times 10^{12} \\\\ & =(2.1 \times 10) \times 10^{12} \\\\ & =2.1 \times 10^{13} \end{aligned}

2. Solve \left(7.2 \times 10^9\right) \times\left(5.6 \times 10^8\right). Write your answer in scientific notation.

4.032 \times 10^{17}

4.032 \times 10^{18}

40.32 \times 10^{17}

0.4032 \times 10^{19}

Use the commutative property to rearrange the expression.

\left(7.2 \times 10^9\right) \times\left(5.6 \times 10^8\right)=(7.2 \times 5.6) \times\left(10^9 \times 10^8\right)

Now solve.

\begin{aligned}& (7.2 \times 5.6) \times\left(10^9 \times 10^8\right) \\\\ & =40.32 \times 10^{17} \end{aligned}

However, 40.32 \times 10^{17} is not in scientific notation since 40.32 is not between 1 and 10.

\begin{aligned}& 40.32 \times 10^{17} \\\\ & =\left(4.032 \times 10^1\right) \times 10^{17} \\\\ & =4.032 \times 10^{18} \end{aligned}

3. Solve \left(3.3 \times 10^4\right) \times\left(6 \times 10^{-7}\right). Write your answer in scientific notation.

2.7 \times 10^{-3}

5.5 \times 10^{11}

1.82 \times 10^{-11}

1.98 \times 10^{-2}

Use the commutative property to rearrange the expression.

\left(3.3 \times 10^4\right) \times\left(6 \times 10^{-7}\right)=(3.3 \times 6) \times\left(10^4 \times 10^{-7}\right)

Now solve.

\begin{aligned}& (3.3 \times 6) \times\left(10^4 \times 10^{-7}\right) \\\\ & =19.8 \times 10^{-3} \end{aligned}

However, 19.8 \times 10^{-3} is not in scientific notation since 19.8 is not between 1 and 10.

\begin{aligned}& 19.8 \times 10^{-3} \\\\ & =\left(1.98 \times 10^1\right) \times 10^{-3} \\\\ & =1.98 \times 10^{-2} \end{aligned}

4. Solve \left(8.6 \times 10^{-5}\right) \times\left(9.35 \times 10^3\right). Write your answer in scientific notation.

80.41 \times 10^{-2}

1.795 \times 10^{-3}

\text { 8. } 041 \times 10^{-1}

7.5 \times 10^{-8}

Use the commutative property to rearrange the expression.

\left(8.6 \times 10^{-5}\right) \times\left(9.35 \times 10^3\right)=(8.6 \times 9.35) \times\left(10^{-5} \times 10^3\right)

Now solve.

\begin{aligned}& (8.6 \times 9.35) \times\left(10^{-5} \times 10^3\right) \\\\ & =80.41 \times 10^{-2} \end{aligned}

However, 80.41 \times 10^{-2} is not in scientific notation since 80.41 is not between 1 and 10.

\begin{aligned}& 80.41 \times 10^{-2} \\\\ & =\left(8.041 \times 10^1\right) \times 10^{-2} \\\\ & =8.041 \times 10^{-1} \end{aligned}

5. A basketball has a circumference of 7.5 \times 10^2 mm. The circumference of the Earth is 5.3 \times 10^7 times larger. What is the circumference of the Earth in mm?

2.2 \times 10^5

1.28 \times 10^{10}

\text { 3. } 975 \times 10^{10}

4 \times 10^8

To solve, multiply 7.5 \times 10^2 by 5.3 \times 10^7.

Use the commutative property to rearrange the expression.

\left(7.5 \times 10^2\right) \times\left(5.3 \times 10^7\right)=(7.5 \times 5.3) \times\left(10^2 \times 10^7\right)

Now solve.

\begin{aligned}& (7.5 \times 5.3) \times\left(10^2 \times 10^7\right) \\\\ & =39.75 \times 10^9 \end{aligned}

However, 39.75 \times 10^{9} is not in scientific notation since 39.75 is not between 1 and 10.

\begin{aligned}& 39.75 \times 10^9 \\\\ & =\left(3.975 \times 10^1\right) \times 10^9 \\\\ & =3.975 \times 10^{10} \end{aligned}

The circumference of the Earth is 3.975 \times 10^{10} mm.

6. A paper clip weighs about 2.2 \times 10^{-3} pounds. How many pounds do 7.21 \times 10^8 paper clips weigh?

1.5862 \times 10^6

5.01 \times 10^5

3.27 \times 10^{11}

3.05 \times 10^{-5}

To solve, multiply 2.2 \times 10^{-3} by 7.21 \times 10^8.

Use the commutative property to rearrange the expression.

\left(2.2 \times 10^{-3}\right) \times\left(7.21 \times 10^8\right)=(2.2 \times 7.21) \times\left(10^{-3} \times 10^8\right)

Now solve.

\begin{aligned}& (2.2 \times 7.21) \times\left(10^{-3} \times 10^8\right) \\\\ & =15.862 \times 10^5 \end{aligned}

However 15.862 \times 10^5 is not in scientific notation since 15.862 is not between 1 and 10.

\begin{aligned}& 15.862 \times 10^5 \\\\ & =\left(1.5862 \times 10^1\right) \times 10^5 \\\\ & =1.5862 \times 10^6 \end{aligned}

The paper clips weigh \text { 1. } 5862 \times 10^6 pounds.

## Scientific notation FAQs

How do you divide in scientific notation?

The rules are similar to the rules for multiplying, except the operation of division is used. You divide numbers in parts; divide the first numbers and the powers of 10 separately.

Then, combine the separate quotients to create the final quote as a power of ten: (quotient of the first numbers) \times (quotient of the powers of 10 ).

What is engineering notation?

It is a number form, similar to scientific notation, but the number is written so that the exponents of 10 are always multiples of 3.

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