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Place value Multiplication and division Integers Exponents Significant figuresHere you will learn about scientific notation including how to convert between ordinary numbers and scientific notation and how to calculate with numbers in scientific notation.

Students will first learn about scientific notation as part of expressions and equations in 8 th grade.

**Scientific notation** is a way of writing very large or very small numbers by using **powers of ten.**

Numbers in scientific notation are written in this format:

a\times10^{n}

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

To write a number in scientific notation, you need to understand the place value of the number.

For example, let’s look at the number 8,290,000 and write the digits in a place value table:

So 8,290,000 written in scientific notation is 8.29\times10^{6} .

**See also:** Powers of 10

Scientific notation is a representation of place value which compliments the decimal number system, as shown in the table below.

Any integer or **terminating decimal** can be written using the scientific notation a\times10^{n}. The table below shows how the value of a can remain the same while the power of ten ‘n’ changes the place value of those digits.

Using scientific notation enables us to write very large or very small numbers.

For example,

67,500,000,000,000,000,000,000=6.75\times10^{22} 0.000000000000037=3.7\times10^{-14}Using scientific notation also enables us to compare the size of very large or very small numbers easily.

For example,

Which is larger: 8,560,000,000,000 or 45,320,000,000,000?

At a glance, it is difficult to tell which is larger, but written in scientific notation, you can compare these numbers very quickly, as shown.

8,560,000,000,000=8.56\times10^{12} 45,320,000,000,000=4.532\times10^{13}Instantly you can see that 4.532\times10^{13} is the larger number as it has the higher power of ten.

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DOWNLOAD FREEWhen numbers are written in scientific notation it can make some calculations neater and quicker to compute.

For example, solve 8.56\times{10^5}-2.3\times{10^2}.

How do you begin? If you expand each number, the expression is 856,000-230.

Notice that none of the digits are in the same place value. So, to solve using scientific notation, convert so that all numbers have the same power of 10.

8.56\times{10^5}=85.6\times{10^4}=856\times{10^3}=8,560\times{10^2}Now the expression is 8,560\times{10^2}-2.3\times{10^2}.

\begin{aligned}&8,560\times{10^2}-2.3\times{10^2} \\\\ &=(8,560-2.3)\times{10^2} \\\\ &=8.557 .7\times{10^2} \end{aligned}Finally, convert 8,557.7\times{10^2} back to scientific notation.

\begin{aligned}&8557.7\times{10^2} \\\\ &=855.77\times{10^3} \\\\ &=85.577\times{10^4} \\\\ &=8.5577\times{10^5} \end{aligned}**Step-by-step guide:** Adding and subtracting scientific notation

Now let’s solve 3.4\times{10^{5}}\times{2}\times{10^{-3}}.

Since multiplication is commutative (the order does not matter), you can reorder this calculation to 3.4\times{2}\times{10^{5}}\times{10^{-3}}.

(3.4\times{2})\times\left({10^5}\times{10^{-3}}\right)Hint: Use the rule a^{b}\times{a^{c}}=a^{b+c} to simplify the powers of 10.

\begin{aligned}&(3.4\times{2})\times\left({10^5}\times{10^{-3}}\right) \\\\ &=6.8\times{10^2}\end{aligned}Since 6.8 is between 1 and 10, you don’t need to adjust the power of 10.

**Step-by-step guide:** How to multiply scientific notation

Now let’s solve \left(3.4\times{10^5}\right)\div\left(2\times{10^3}\right).

Re–write this expression as \cfrac{3.4\times{10^5}}{2\times{10^3}}, which equals \cfrac{3.4\times{10}\times{10}\times{10}\times{10}\times{10}}{2\times{10}\times{10}\times{10}}.

Notice how you can divide the corresponding parts to simplify.

This is the same as solving:

\begin{aligned}&\left(3.4\times{10^5}\right)\div\left(2\times{10^3}\right) \\\\ &=(3.4\div{2})\times\left({10^5}\div{10^3}\right) \\\\ &=1.7\times{10^2}\end{aligned}Since 1.7 is between 1 and 10, you don’t need to adjust the power of 10.

**Step-by-step guide:** How to divide scientific notation

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 (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

In order to represent a number in scientific notation:

**Show the number as an expression with multipliers of \bf{10}.****Determine the power of \bf{10}.****Write the number.**

Write this number in scientific notation: 52,000

**Show the number as an expression with multipliers of \bf{10}.**

Remember the first number is always between 1 and 10 .

2**Determine the power of \bf{10}. **

3**Write the equation.**

In order to solve problems with numbers written in scientific notation:

**Use place value reasoning to identify how the power of \bf{10} will change the number.****Shift the digits left if multiplying or right if dividing.**

Write 9.4\times10^{5} as an ordinary number.

**Use place value reasoning to identify how the power of \bf{10} will change the number.**

9.4\times{10^5}=9.4\times{10}\times{10}\times{10}\times{10}\times{10}=9.4\times100,000

Each digit must move 5 places to the left.

**Shift the digits left if multiplying or right if dividing.**

9.4\times{10^5}=940000

In order to add and subtract numbers in scientific notation:

**Convert the number(s) to have the same power of \bf{10}.****Add or subtract the non-zero digits.****Check your answer is in scientific notation.**

Calculate 6\times10^{4}+2\times10^{3}. Write your answer in scientific notation.

**Convert the number(s) to have the same power of \bf{10}. **

Let’s convert 2\times10^{3} to be 10^4.

To do this, multiply it by 10 to add one to the power. To maintain the value of the number you need to divide the non-zero number by 10.

**Add or subtract the non-zero digits.**

\begin{aligned}&6\times{10^4}+0.2\times{10^4} \\\\
&=(6+0.2)\times{10^4} \\\\
&=6.2\times{10^4} \end{aligned}

**Check your answer is in scientific notation.**

Since 6.2 is between 1 and 10, you don’t need to adjust the power of 10.

Calculate 5\times{10^8}-4\times{10^6}. Write your answer in scientific notation.

**Convert the number(s) to have the same power of \bf{10}. **

Let’s convert 5\times{10^8} to be 10^6.

5\times{10^8}=50\times{10^7}=500\times{10^6}

**Add or subtract the non-zero digits.**

\begin{aligned}&500\times{10^6}-4\times{10^6} \\\\
&=(500-4)\times{10^6} \\\\
&=496\times{10^6} \end{aligned}

**Check your answer is in scientific notation.**

496 is not between 1 and 10.

Convert 496\times{10^6} back to scientific notation.

In order to multiply and divide in scientific notation:

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

Calculate \left(5\times{10^4}\right)\times\left(7\times{10^8}\right).

Write your answer in scientific notation.

**Multiply or divide the non-zero numbers.**

5\times{7}=35

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

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

**Write the solution in scientific notation.**

35\times{10^{12}}

35 is not between 1 and 10.

Convert 35\times{10^{12}} back to scientific notation.

\left(5\times{10^4}\right)\times\left(7\times{10^8}\right)=3.5\times{10^{13}}

Calculate 8\times{10^7}\div{2}\times{10^2}. Write your answer in scientific notation.

**Multiply or divide the non-zero numbers.**

8\div{2}=4

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

{10^7}\div{10^2}={10^{7-2}}={10^5}

**Write the solution in scientific notation.**

4\times{10^5}

- Give students real world examples of when scientific notation is useful, such as the distance from the sun to Neptune or the size of a microorganism.

- Start with smaller numbers when students are first learning. This allows students an opportunity to make sense of problems or check their work by expanding numbers written in scientific notation, until they understand the rules for operating.

- Allow struggling students to use a scientific notation calculator.

**Writing a number with the incorrect exponent**

This can happen by counting all the zeros for large numbers or all zeros after the decimal point for small numbers, then writing this as the exponent, without considering the other digits in the number or their place value.

For example,

**Not converting numbers to the same power of \bf{10} when adding or subtracting**

Multiplying by different powers of 10 means the first part of the number does not represent the same place value. In order to use the subtraction algorithm, the first number needs to represent the same place value positions.

For example, 5.6\times{10^3}+6.7\times{10^2}

5.6\times{10^3} is read as “ 5.6 thousands” and 6.7\times{10^2} is read as “ 6.7 hundreds.”

In order to add them, they need to represent the same place value.

Since, 5.6\times{10^3}=56\times{10^2}, you can solve 56\times{10^2}+6.7\times{10^2}

**Not converting solutions to scientific notation**

After calculating with scientific notation, a common mistake is to not check that the first part of the number is 1\leq{n}<10.

**Forgetting the meaning of negative exponents**

Remember that decimals are represented as smaller powers of 10. The negative exponent represents the equivalent fractional value and is used for every position to the right of the decimal.

For example,

10^{-1}=\cfrac{1}{10}

10^{-2}=\cfrac{1}{100}

10^{-3}=\cfrac{1}{1,000}

10^{-4}=\cfrac{1}{10,000}

1. Write 86,000 in scientific notation.

8.6\times{10^4}

8.6\times{10^3}

86\times{10^3}

8.6\times{10^{-4}}

The number between 1 and 10 here is 8.6. Since 8 is in the ten thousands column,

\begin{aligned}86,000&=8.6\times{10,000} \\\\ &=8.6\times{10^4}\end{aligned}

2. Write 0.0097 in scientific notation.

9.7\times{10^3}

9.7\times{10^{-3}}

9.7\times{10^{-4}}

0.97\times{10^{-2}}

The number between 1 and 10 here is 9.7. Since 9 is in the thousandths position

\begin{aligned}0.0097&=9.7\times\cfrac{1}{1,000} \\\\ &=9.7\times{10^{-3}}\end {aligned}

3. Write 5.9\times{10^3} as an ordinary number.

59,000

590

5,900

0.0059

10^{3}=1,000 therefore

\begin{aligned}5.9\times{10^3}&=5.9\times{1,000} \\\\ &=5,900\end{aligned}

4. Solve 7\times{10^5}+2\times{10^4}. Write your answer in scientific notation.

7.2 \times{10^5}

7.2\times{10^4}

72\times{10^4}

9\times{10^9}

Before adding, the first number needs to have the same place values. To do this, both powers of 10 need to be the same.

Since 2\times{10^4}=0.2\times{10^5}, solve 7\times{10^5}+0.2\times{10^5}.

\begin{aligned}&7\times{10^5}+0.2\times{10^5} \\\\ &=(7+0.2)\times{10^5} \\\\ &=7.2\times{10^5} \end{aligned}

5. Solve \left(6\times{10^8}\right)\times\left(3\times{10^4}\right). Write your answer in scientific notation.

18\times{10^{12}}

18\times{10^{32}}

1.8\times{10^{11}}

1.8\times{10^{13}}

Use the commutative property to rearrange the expression.

\left(6\times{10^8}\right)\times\left(3\times{10^4}\right)=(6\times{3}) \times\left({10^8}\times{10^4}\right)

Now solve.

\begin{aligned}&(6\times{3})\times\left({10^8}\times{10^4}\right) \\\\ &=18\times{10^{12}}\end{aligned}

However 18\times{10^{12}} is not in scientific notation since 18 is not between 1 and 10.

\begin{aligned}&18\times{10^{12}} \\\\ &=(1.8\times{10})\times{10^{12}} \\\\ &=1.8\times{10^{13}} \end{aligned}

6. Solve \left(9\times{10^7}\right)\div(4\times{10^2}). Write your answer in scientific notation.

2.25\times{10^5}

22.5\times{10^4}

0.225\times{10^6}

2.25\times{10^3.5}

Re–write this expression as \cfrac{9\times{10^7}}{4\times{10^2}},

which equals \cfrac{9\times{10}\times{10}\times{10}\times{10}\times{10}\times{10}\times{10}}{4\times{10}\times{10}}.

Notice how you can divide the corresponding parts to simplify.

This is the same as solving:

\begin{aligned}&\left(9\times{10^7}\right)\div\left(4\times{10^2}\right) \\\\ &=(9\div{4})\times\left({10^7}\div{10^2}\right) \\\\ &=2.25\times{10^5}\end{aligned}

Yes. For example, Richard, David and James did a survey, asking people what the best genre of movie was.

If 1.3\times{10^4} people said sci-fi and 4\times{10^2} more people said historical fiction, how many people said historical fiction?

This word problem can be solved by adding the numbers written in scientific notation.

This and other terms, such as ‘standard index form’ or ‘standard form’ (in the UK), all have the same meaning as scientific notation.

These are the digits used to express a number to the desired form of accuracy.

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

Each power of 10 has a prefix in the metric system, since the system is based on powers of 10.

The distance a number is from 0.

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