Operations in Scientific Notation

Q: Why do I add exponents when multiplying?

It is the product rule for powers: $10^{3} \cdot 10^{4}$ is three factors of $10$ times four more factors of $10$, which is seven factors in all — $10^{7}$. Multiplying the powers of $10$ stacks their factors, so the exponents add.

Q: What do I do if the front number comes out 10 or bigger?

Move the decimal one place left and add $1$ to the exponent: $20 \cdot 10^{9} = 2 \cdot 10^{10}$. If the front number comes out smaller than $1$, move the decimal right and subtract $1$ instead: $0.4 \cdot 10^{6} = 4 \cdot 10^{5}$.

Q: Can I add two numbers with different powers of 10?

Yes, but not directly — rewrite one of them so the exponents match first. $4 \cdot 10^{5} + 3 \cdot 10^{4}$ becomes $4 \cdot 10^{5} + 0.3 \cdot 10^{5} = 4.3 \cdot 10^{5}$. Matching exponents plays the same role as a common denominator.

Q: Do the exponents ever get multiplied?

Not in these operations. Exponents multiply only when a power is raised to another power, like $(10^{3})^{4} = 10^{12}$. Multiplying two numbers in scientific notation adds exponents; dividing subtracts them.

A number in scientific notation has the form $a \cdot 10^{n}$ , where the front number $a$ satisfies $1 \leq a < 10$ . The payoff of writing numbers this way is that arithmetic gets easy: you handle the front numbers and the powers of $10$ as two separate, small jobs instead of dragging long strings of zeros through a calculation.

There are really only two procedures to learn. To multiply or divide, work the front numbers and the exponents separately. To add or subtract, make the exponents match first, then combine the front numbers. Everything else in this lesson is cleanup.

Multiplying and dividing

To multiply: multiply the front numbers, then add the exponents on the powers of $10$ . So $(2 \cdot 10^{3})(3 \cdot 10^{4}) = 6 \cdot 10^{7}$ , because $2 \cdot 3 = 6$ and $10^{3} \cdot 10^{4} = 10^{7}$ .

To divide: divide the front numbers, then subtract the exponents. So $\dfrac{9 \cdot 10^{7}}{3 \cdot 10^{2}} = 3 \cdot 10^{5}$ , because $9 \div 3 = 3$ and $\dfrac{10^{7}}{10^{2}} = 10^{5}$ . These are just the product and quotient rules for exponents applied to a base of $10$ .

When the front number breaks the rules

After multiplying or dividing, check the front number. Scientific notation requires $1 \leq a < 10$ , and the arithmetic does not always cooperate. $(4 \cdot 10^{3})(5 \cdot 10^{6})$ gives $20 \cdot 10^{9}$ — and $20$ is too big to be a legal front number.

The fix is one decimal move: $20 = 2 \cdot 10$ , so $20 \cdot 10^{9} = 2 \cdot 10^{10}$ . Moving the decimal one place left adds $1$ to the exponent; moving it one place right subtracts $1$ . Always do this check before circling an answer — the too-big front number is the most common trap on this topic.

Adding and subtracting

Addition and subtraction have one extra requirement: the exponents must match first. Once they do, add or subtract the front numbers and keep the power of $10$ : $3.1 \cdot 10^{4} + 2.5 \cdot 10^{4} = 5.6 \cdot 10^{4}$ .

If the exponents do not match, rewrite one number so they do. To add $4 \cdot 10^{5} + 3 \cdot 10^{4}$ , rewrite $3 \cdot 10^{4}$ as $0.3 \cdot 10^{5}$ , then add: $4.3 \cdot 10^{5}$ . Matching exponents means both numbers are counted in the same units, the same way you need common denominators before adding fractions.

Worked examples

Example 1: multiply

Multiply $(2 \cdot 10^{3})(4 \cdot 10^{5})$ . Write the answer in scientific notation.

Multiply the front numbers

2 \cdot 4 = 8

Add the exponents

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

Check the front number:

1 \leq 8 < 10

✓

8 \cdot 10^{8}

Answer: $8 \cdot 10^{8}$

Example 2: divide

Divide $\dfrac{9 \cdot 10^{7}}{3 \cdot 10^{2}}$ . Write the answer in scientific notation.

Divide the front numbers

9 \div 3 = 3

Subtract the exponents

\dfrac{10^{7}}{10^{2}} = 10^{5}

Combine

3 \cdot 10^{5}

Answer: $3 \cdot 10^{5}$

Example 3: multiply, then fix the front number

Multiply $(6 \cdot 10^{2})(5 \cdot 10^{4})$ . Write the answer in scientific notation.

Multiply the front numbers

6 \cdot 5 = 30

Add the exponents

10^{2} \cdot 10^{4} = 10^{6}

30

is too big for a front number

30 \cdot 10^{6}

Move the decimal left, bump the exponent

30 \cdot 10^{6} = 3 \cdot 10^{7}

Answer: $3 \cdot 10^{7}$

Try one yourself

Common questions

Why do I add exponents when multiplying?

It is the product rule for powers: $10^{3} \cdot 10^{4}$ is three factors of $10$ times four more factors of $10$ , which is seven factors in all — $10^{7}$ . Multiplying the powers of $10$ stacks their factors, so the exponents add.

What do I do if the front number comes out 10 or bigger?

Move the decimal one place left and add $1$ to the exponent: $20 \cdot 10^{9} = 2 \cdot 10^{10}$ . If the front number comes out smaller than $1$ , move the decimal right and subtract $1$ instead: $0.4 \cdot 10^{6} = 4 \cdot 10^{5}$ .

Can I add two numbers with different powers of 10?

Yes, but not directly — rewrite one of them so the exponents match first. $4 \cdot 10^{5} + 3 \cdot 10^{4}$ becomes $4 \cdot 10^{5} + 0.3 \cdot 10^{5} = 4.3 \cdot 10^{5}$ . Matching exponents plays the same role as a common denominator.

Do the exponents ever get multiplied?

Not in these operations. Exponents multiply only when a power is raised to another power, like $(10^{3})^{4} = 10^{12}$ . Multiplying two numbers in scientific notation adds exponents; dividing subtracts them.

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Operations in Scientific Notation

Multiplying and dividing

When the front number breaks the rules

Adding and subtracting

Worked examples

Example 1: multiply

Example 2: divide

Example 3: multiply, then fix the front number

Try one yourself

Common questions

Keep learning

Writing Numbers in Scientific Notation

Power of a Power

Zero & Negative Exponents

Exponent Notation & Meaning