Math Core

Scientists don't just write huge and tiny numbers, they calculate with them: how far light travels in a year, how many cells are in a body, how much data fits on a drive. The properties of exponents make these calculations quick, because you can work with the coefficients and the powers of 10 separately.

Multiplying

Multiplication can be done in any order, so you can group the coefficients together and the powers of 10 together:

(3 \times 10^4)(2 \times 10^5) = (3 \times 2) \times (10^4 \times 10^5) = 6 \times 10^9

Multiply the coefficients, and add the exponents of 10.

Sometimes the new coefficient is 10 or more. Then you need one more step to put the answer back in scientific notation.

Worked example: Multiplying and adjusting

Find $(5 \times 10^3)(4 \times 10^6)$ . Write the answer in scientific notation.

\begin{aligned} (5 \times 10^3)(4 \times 10^6) &= (5 \times 4) \times 10^{3+6} \\ &= 20 \times 10^9 \end{aligned}

But $20$ is not between 1 and 10. Rewrite it as $2 \times 10^1$ :

20 \times 10^9 = 2 \times 10^1 \times 10^9 = 2 \times 10^{10}

Adjusting the coefficient

If the coefficient is 10 or more, move its decimal point left one place and add 1 to the exponent: $20 \times 10^9 = 2 \times 10^{10}$ .
If the coefficient is less than 1, move its decimal point right one place and subtract 1 from the exponent: $0.3 \times 10^6 = 3 \times 10^5$ .

The value of the number doesn't change. You make the coefficient smaller and the power of 10 larger (or the other way around) by the same factor of 10.

Dividing

Division works the same way. Divide the coefficients, and subtract the exponents of 10.

Worked example: Dividing

Find $\dfrac{8.4 \times 10^9}{2.1 \times 10^3}$ .
Find $\dfrac{1.5 \times 10^4}{5 \times 10^{-2}}$ .

Solutions.

$\dfrac{8.4}{2.1} = 4$ and $10^{9-3} = 10^6$ , so the quotient is $4 \times 10^6$ .
$\dfrac{1.5}{5} = 0.3$ and $10^{4-(-2)} = 10^6$ , so the quotient is $0.3 \times 10^6$ . The coefficient is less than 1, so adjust: $0.3 \times 10^6 = 3 \times 10^5$ .

Common mistake

Be careful subtracting a negative exponent: $10^4 \div 10^{-2} = 10^{4-(-2)} = 10^{6}$ , not $10^2$ . Write the subtraction out with parentheses.

Adding and subtracting

Adding is different. You cannot add the exponents. Instead, the numbers must have the same power of 10, just like you can only add fractions with a common denominator. Then you add or subtract the coefficients and keep the power of 10.

Worked example: Adding with different exponents

Find $(3.2 \times 10^5) + (4.1 \times 10^4)$ .

Rewrite the second number so it also uses $10^5$ . Since $4.1 \times 10^4 = 41{,}000 = 0.41 \times 10^5$ :

\begin{aligned} (3.2 \times 10^5) + (0.41 \times 10^5) &= (3.2 + 0.41) \times 10^5 \\ &= 3.61 \times 10^5 \end{aligned}

Check in standard form: $320{,}000 + 41{,}000 = 361{,}000 = 3.61 \times 10^5$ .

Tip

If adding feels confusing, write both numbers in standard form, add, and convert back. Scientific notation is a tool, not a rule you must stay inside.

A real-world problem

Worked example: How far does sunlight travel?

Light travels about $3 \times 10^5$ kilometers per second. Sunlight takes about $5 \times 10^2$ seconds to reach Earth. About how far away is the Sun?

Distance equals speed times time:

\begin{aligned} (3 \times 10^5)(5 \times 10^2) &= 15 \times 10^7 \\ &= 1.5 \times 10^8 \end{aligned}

The Sun is about $1.5 \times 10^8$ kilometers, or 150,000,000 kilometers, away.

Calculators often show scientific notation with an E. A display of 1.5E8 means $1.5 \times 10^8$ , and 2.4E-6 means $2.4 \times 10^{-6}$ .