Scientific Notation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Scientific Notation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A way of writing very large or very small numbers as a×10na \times 10^n, where 1≤∣a∣<101 \leq |a| < 10 and nn is an integer.

Instead of writing out all the zeros in 93,000,000 or 0.000042, you slide the decimal point and count how many places it moved. The exponent on 10 keeps track of the shift.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Scientific notation separates a number into a size between 1 and 10 and a power of ten.

Common stuck point: The procedure for scientific notation is the easy part; the trap is choosing a first factor outside 1≤a<101\le a<10. Asking "Is the first factor at least 1 and less than 10?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the first factor at least 1 and less than 10?

Worked Examples

Example 1

easy
Write 0.000470.00047 in scientific notation.

Answer

4.7×10−44.7 \times 10^{-4}

First step

1
Move the decimal point right until we have a number between 1 and 10: 4.74.7.

Full solution

  1. 2
    Count the places moved: 4 places to the right, so the exponent is −4-4.
  2. 3
    Result: 4.7×10−44.7 \times 10^{-4}.
Scientific notation expresses a number as a×10na \times 10^n where 1≤a<101 \leq a < 10. Moving the decimal right gives a negative exponent; moving it left gives a positive exponent.

Example 2

medium
Compute (3.0×105)×(2.0×10−3)(3.0 \times 10^5) \times (2.0 \times 10^{-3}) and express the answer in scientific notation.

Example 3

easy
Write 1 million in scientific notation.

Example 4

medium
A virus is 1.2×10−71.2 \times 10^{-7} m wide. Express this in standard form.

Example 5

hard
Earth's mass is about 6×10246 \times 10^{24} kg, Moon's mass is about 7.3×10227.3 \times 10^{22} kg. How many times more massive is Earth than the Moon?

Example 6

hard
Compute (2×10−4)3(2 \times 10^{-4})^3 in scientific notation.

Example 7

challenge
The mass of one atom of carbon is about 1.99×10−231.99 \times 10^{-23} g. About how many atoms are in 12 g of carbon? Express in scientific notation.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Convert 6.02×10236.02 \times 10^{23} to standard form and identify what famous constant this represents.

Example 2

easy
Write 45,000,00045{,}000{,}000 in scientific notation.

Example 3

easy
Write 93,000,00093{,}000{,}000 in scientific notation.

Example 4

easy
Write 0.0000420.000042 in scientific notation.

Example 5

easy
Convert 5×1035\times10^3 to standard form.

Example 6

easy
Convert 7×10−47\times10^{-4} to standard form.

Example 7

easy
Is 12×10412\times10^4 in proper scientific notation? If not, fix it.

Example 8

easy
Write 400400 in scientific notation.

Example 9

easy
Convert 6.5×1006.5\times10^0 to standard form.

Example 10

easy
Which is larger: 3×1053\times10^5 or 9×1049\times10^4?

Example 11

medium
Multiply (2×103)(4×105)(2\times10^3)(4\times10^5).

Example 12

medium
Divide 8×1062×102\frac{8\times10^6}{2\times10^2}.

Example 13

medium
Multiply (5×103)(6×104)(5\times10^3)(6\times10^4) and put the result in proper form.

Example 14

medium
Add 3×104+5×1043\times10^4+5\times10^4.

Example 15

medium
Add 4×105+3×1044\times10^5+3\times10^4.

Example 16

medium
The mass of a proton is about 1.6×10−271.6\times10^{-27} kg. Write this in standard form.

Example 17

medium
Express 0.00056×1030.00056\times10^3 in proper scientific notation.

Example 18

challenge
Light travels 3×1083\times10^8 m/s. How far in 2×1022\times10^2 seconds? Give scientific notation.

Example 19

challenge
Estimate how many times larger 6×10126\times10^{12} is than 3×1043\times10^4.

Example 20

challenge
Simplify (4×105)(3×10−2)6×101\frac{(4\times10^5)(3\times10^{-2})}{6\times10^1} in scientific notation.

Example 21

medium
Write 0.0036×1050.0036\times10^5 in proper scientific notation.

Example 22

medium
Square (3×104)2(3\times10^4)^2 and give scientific notation.

Example 23

easy
Write 250,000250{,}000 in scientific notation.

Example 24

easy
Write 0.000380.00038 in scientific notation.

Example 25

easy
Is 0.45×1060.45 \times 10^6 in proper scientific notation? If not, fix it.

Example 26

medium
Compute (4×103)(2×105)(4 \times 10^3)(2 \times 10^5) in scientific notation.

Example 27

medium
Compute 6×1093×104\frac{6 \times 10^9}{3 \times 10^4} in scientific notation.

Example 28

medium
The speed of light is about 3×1083 \times 10^8 m/s. Express this in standard form.

Example 29

medium
Add 3.2×105+4.5×1053.2 \times 10^5 + 4.5 \times 10^5 and express in scientific notation.

Example 30

medium
Write 670×104670 \times 10^4 in proper scientific notation.

Example 31

medium
Compute (5×10−3)(4×107)(5 \times 10^{-3})(4 \times 10^7) in scientific notation.

Example 32

medium
Subtract: 7.5×104−2.1×1047.5 \times 10^4 - 2.1 \times 10^4 in scientific notation.

Example 33

hard
Add 3.0×105+4.0×1043.0 \times 10^5 + 4.0 \times 10^4 in scientific notation.

Example 34

hard
Compute (2.5×106)2(2.5 \times 10^6)^2 in scientific notation.

Example 35

hard
Express 0.0005060.000506 in scientific notation with proper form.

Example 36

hard
A computer performs 5×1095 \times 10^9 operations per second. How many operations does it do in 2×1032 \times 10^3 seconds?

Example 37

hard
Compute 9×1016\sqrt{9 \times 10^{16}} in scientific notation.

Example 38

challenge
A bacterial culture starts with 2×1042 \times 10^4 cells and doubles every hour. How many cells after 10 hours? Express in scientific notation.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

exponent rulesplace valuedecimals