Writing Numbers in Scientific Notation

Q: Is $52 \times 10^{3}$ scientific notation?

No. It equals $52{,}000$, but the front number in scientific notation must be at least $1$ and less than $10$. Rewrite $52$ as $5.2 \times 10^{1}$, combine the powers of $10$, and you get the correct form: $5.2 \times 10^{4}$.

Scientific notation is a short way to write very big or very small numbers. Instead of writing $150{,}000{,}000$ , you write $1.5 \times 10^{8}$ — a small decimal times a power of $10$ . Scientists use it for things like the distance to the Sun or the width of a cell, but it shows up on every pre-algebra test too.

Every number in scientific notation has the same shape: a number that is at least $1$ but less than $10$ , multiplied by a power of $10$ . Once you can move a decimal point and count the places, you can write any number this way — and read any number written this way.

The rule: one digit in front

A number is in scientific notation when it looks like $a \times 10^{n}$ , where $a$ is at least $1$ and less than $10$ . That means exactly one nonzero digit sits in front of the decimal point. $5.2 \times 10^{4}$ is scientific notation; $52 \times 10^{3}$ is not, even though it equals the same amount, because $52$ is too big to be the front number.

The exponent $n$ tells you how many places the decimal point moved. A big number (more than $10$ ) gets a positive exponent. A small number (less than $1$ ) gets a negative exponent. You may also see the multiplication written with a raised dot, as in $5.2 \cdot 10^{4}$ — it means exactly the same thing as $5.2 \times 10^{4}$ .

Converting in both directions

Standard form to scientific notation: place the decimal point right after the first nonzero digit, then count how many places it moved from where it started. That count is the exponent. For $52{,}000$ , the decimal moves $4$ places to sit after the $5$ , so you get $5.2 \times 10^{4}$ . For $0.00087$ , it moves $4$ places the other way, so you get $8.7 \times 10^{-4}$ .

Scientific notation to standard form: reverse the process. A positive exponent moves the decimal to the right, making the number bigger; a negative exponent moves it to the left, making the number smaller. Fill empty places with zeros.

A quick sanity check: if the original number was huge, the exponent must be positive. If it was a tiny decimal, the exponent must be negative. Mixing those up is the most common mistake on this topic.

Multiplying numbers in scientific notation

To multiply two numbers in scientific notation, multiply the front numbers together and add the exponents on the $10$ s (that is the product rule for exponents: $10^{a} \times 10^{b} = 10^{a+b}$ ).

One catch: if the new front number comes out $10$ or bigger, the answer is not in scientific notation yet. Rewrite the front number and bump the exponent up by one. Example 4 below shows exactly that.

Worked examples

Example 1: a big number into scientific notation

Write $4{,}500{,}000$ in scientific notation.

Place the decimal after the first nonzero digit

4.5

Count the places the decimal moved

4{,}500{,}000 \rightarrow 4.5 \text{ is } 6 \text{ places}

Big number, so the exponent is positive

4{,}500{,}000 = 4.5 \times 10^{6}

Answer: $4.5 \times 10^{6}$

Example 2: a small number into scientific notation

Write $0.00032$ in scientific notation.

Place the decimal after the first nonzero digit

3.2

Count the places the decimal moved

0.00032 \rightarrow 3.2 \text{ is } 4 \text{ places}

Small number, so the exponent is negative

0.00032 = 3.2 \times 10^{-4}

Answer: $3.2 \times 10^{-4}$

Example 3: back to standard form

Write $7.04 \times 10^{5}$ in standard form.

The exponent is

+5

, so move the decimal

5

places to the right

7.04 \rightarrow 704{,}000

Fill the empty places with zeros

7.04 \times 10^{5} = 704{,}000

Answer: $704{,}000$

Example 4: multiplying, with an adjustment

Multiply $(4 \times 10^{5})(5 \times 10^{3})$ and write the answer in scientific notation.

Multiply the front numbers

4 \times 5 = 20

Add the exponents

10^{5} \times 10^{3} = 10^{8}

So far

20 \times 10^{8}

20

is too big to be the front number — rewrite it

20 = 2 \times 10^{1}

Bump the exponent up by one

2 \times 10^{9}

Answer: $2 \times 10^{9}$

Try one yourself

Common questions

How do I know if the exponent should be positive or negative?

Look at the original number. If it is $10$ or bigger, the exponent is positive. If it is smaller than $1$ , the exponent is negative. The exponent just records which way (and how far) the decimal point moved.

52 \times 10^{3}

scientific notation?

No. It equals $52{,}000$ , but the front number in scientific notation must be at least $1$ and less than $10$ . Rewrite $52$ as $5.2 \times 10^{1}$ , combine the powers of $10$ , and you get the correct form: $5.2 \times 10^{4}$ .

What does a negative exponent mean here?

It means the number is a small decimal, not a negative number. $8.7 \times 10^{-4} = 0.00087$ , which is positive. The negative sign only tells you the decimal point moved to the left.

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