Why do we subtract the exponents?

Because $10^m$ is $m$ tens multiplied together. Dividing by $10^n$ cancels $n$ of those tens: $\frac{10^5}{10^2} = \frac{10 \cdot 10 \cdot 10 \cdot 10 \cdot 10}{10 \cdot 10} = 10^3$. That is the quotient rule $10^m \div 10^n = 10^{m-n}$.

What if both numbers have the same exponent?

Then the exponents cancel to $10^0 = 1$ and only the coefficients matter: $(9 \times 10^4) \div (3 \times 10^4) = 3 \times 10^0 = 3$.

How many significant figures should I keep?

Usually as many as the least precise number in the problem. If the data have 2 or 3 significant figures, round the coefficient to 2 or 3 as well. Keeping more digits than the data have gives a false sense of accuracy.

Teacher Resources

Dividing in Scientific Notation

Divide numbers in scientific notation: divide the coefficients, subtract the exponents, then normalize the result so the coefficient is between 1 and 10.

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Definition

To divide numbers in scientific notation, split the problem into two easy parts: divide the coefficients, and divide the powers of ten by subtracting the exponents (the quotient rule

10^m \div 10^n = 10^{m-n}

\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}

Then check the coefficient. In scientific notation it must satisfy

1 \le a < 10

. When you divide, the new coefficient can come out smaller than 1, so you normalize it: move the decimal point one place to the right (multiply the coefficient by

10

) and subtract

1

from the exponent.

\frac{1.2 \times 10^5}{3 \times 10^2} = 0.4 \times 10^3 = 4 \times 10^2

Two more things to watch:

Negative exponents: subtracting a negative exponent adds, so $10^{-4} \div 10^{-7} = 10^{-4-(-7)} = 10^3$ .
Quotients that never end: $7 \div 3 = 2.333\ldots$ , so round the coefficient, for example to $3$ significant figures: $2.33$ .

Try it now

Calculate

(6 \times 10^8) \div (2 \times 10^3)

Worked Examples

Calculate

(8 \times 10^9) \div (2 \times 10^3)

Divide the coefficients

8 \div 2 = 4

Subtract the exponents (quotient rule)

10^9 \div 10^3 = 10^{9-3} = 10^6

Combine and check the coefficient

4 \times 10^6

, and

1 \le 4 < 10

→ Already normalized

Answer: $(8 \times 10^9) \div (2 \times 10^3) = 4 \times 10^6$

Common Mistakes

Dividing the exponents: $10^8 \div 10^2 = 10^4$

Why it's wrong: The exponent counts how many tens are multiplied together. Dividing cancels tens, so you take away the exponent, you do not divide it.

Correct: Subtract: $10^8 \div 10^2 = 10^{8-2} = 10^6$ (eight tens divided by two tens leaves six tens).

Subtracting in the wrong order: $10^3 \div 10^7 = 10^4$

Why it's wrong: Order matters in division. The exponent of the number you divide BY is subtracted from the exponent on top.

Correct: Always top minus bottom: $10^3 \div 10^7 = 10^{3-7} = 10^{-4}$ , a small number, as expected when you divide by something bigger.

Normalizing in the wrong direction: $0.4 \times 10^3 = 4 \times 10^4$

Why it's wrong: Making the coefficient 10 times bigger means the power of ten must get 10 times smaller, otherwise the value changes from $400$ to $40{,}000$ .

Correct: Coefficient up by one place, exponent down by one: $0.4 \times 10^3 = 4 \times 10^2$ . Check: both equal $400$ .

Losing a sign with negative exponents: $10^5 \div 10^{-3} = 10^2$

Why it's wrong: The second minus sign was dropped. Subtracting $-3$ is the same as adding $3$ .

Correct: $10^5 \div 10^{-3} = 10^{5-(-3)} = 10^{5+3} = 10^8$ . Dividing by a tiny number gives a big result.

Interactive Visual

Divide, subtract the exponents, then normalize 0.4 × 10³

Divide

(1.2 \times 10^{5}) \div (3 \times 10^{2})

1Group the coefficients and the powers of 10.
$\dfrac{1.2}{3} \times \dfrac{10^{5}}{10^{2}}$

Divide the coefficients. What do you get?

A quotient that never ends: round to 3 significant figures

Divide

(7 \times 10^{8}) \div (3 \times 10^{3})

1Group the coefficients and the powers of 10.
$\dfrac{7}{3} \times \dfrac{10^{8}}{10^{3}}$

Divide the coefficients. What do you get?

Negative exponents: subtracting a negative adds

Divide

(6 \times 10^{-4}) \div (2 \times 10^{-7})

1Group the coefficients and the powers of 10.
$\dfrac{6}{2} \times \dfrac{10^{-4}}{10^{-7}}$

Divide the coefficients. What do you get?

Coefficient a6Exponent m−4Coefficient b2Exponent n−7

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Practice Problems

18 problems

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Easy

Calculate

(6 \times 10^8) \div (2 \times 10^3)

Why It Matters

Division answers questions like "how long?", "how many per ...?" and "how much each?". In science and in the news, the numbers in those questions are often huge or tiny:

Light from the Sun: the Sun is about $1.5 \times 10^{11}$ m away and light travels $3 \times 10^8$ m every second. Dividing tells you sunlight is about $500$ seconds old when it reaches you.
Population density: a country's population divided by its area gives people per km², the number geographers use to compare countries.
Per-person share: a national budget or debt divided by the population tells you what it means for one person.
Biology and chemistry: dividing the mass of a water drop by the mass of one molecule tells you how many molecules are in it (about $10^{21}$ ).

Writing these numbers out in full would mean counting long rows of zeros. In scientific notation you divide two small numbers and subtract two exponents, and you are done.

Real World Applications

Astronomy: Travel Time of Light

Astronomers divide a distance by the speed of light to find how long light has been travelling. That is why we see the Sun as it was a few minutes ago and distant stars as they were years ago.

Example:

The Moon is about $3.84 \times 10^8$ m away. Light needs $(3.84 \times 10^8) \div (3 \times 10^8) = 1.28 \times 10^0 = 1.28$ seconds to get there.

1Try It Yourself

The Sun is about $1.5 \times 10^{11}$ m from Earth, and light travels $3 \times 10^8$ m per second.

How many seconds does light take to travel from the Sun to Earth?

Step 1: Write the mathematical expression

Time = distance ÷ speed:

Geography: Population Density

Population density is the number of people divided by the land area. It shows how crowded a country or city is, and it is how planners compare places of very different sizes.

Example:

Germany has about $8.4 \times 10^7$ people on about $3.5 \times 10^5$ km². Density: $(8.4 \div 3.5) \times 10^{7-5} = 2.4 \times 10^2$ , about $240$ people per km².

2Try It Yourself

A large city has $3.6 \times 10^6$ inhabitants living on $9 \times 10^2$ km².

How many people live on one square kilometre?

Step 1: Write the mathematical expression

Density = people ÷ area:

Economics: Per-Person Share

News reports give national figures (budgets, debts, total spending) in billions or trillions. Dividing by the population turns them into an amount per person that is easy to imagine.

Example:

A country of $4 \times 10^7$ people spends $1.2 \times 10^{11}$ dollars a year on food: $(1.2 \div 4) \times 10^{11-7} = 0.3 \times 10^4 = 3 \times 10^3$ , so about 3000 dollars per person.

3Try It Yourself

A country's national debt is $2.4 \times 10^{12}$ dollars, and $3 \times 10^8$ people live there.

What is each person's share of the debt, in dollars?

Step 1: Write the mathematical expression

Share = debt ÷ people:

Key Takeaways

1Divide the coefficients and divide the powers of ten separately: $\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}$
2Quotient rule: subtract the exponents, always top minus bottom ( $10^3 \div 10^7 = 10^{-4}$ )
3If the new coefficient is below 1, normalize: coefficient one place right, exponent minus 1 ( $0.4 \times 10^3 = 4 \times 10^2$ )
4Subtracting a negative exponent adds: $10^{-4} \div 10^{-7} = 10^{3}$
5When the coefficients do not divide evenly, round the coefficient (for example to 3 significant figures) and write $\approx$
6Division answers "how long", "how many per" and "how much each" questions: light travel time, population density, per-person shares

Frequently Asked Questions

No. Both coefficients are between 1 and 10, so their quotient is always more than

0.1

and less than

10

. That is why, when dividing, you only ever need to normalize in one direction: a coefficient below 1 moves one place to the right and the exponent goes down by 1.

Glossary

Coefficient: The number $a$ in $a \times 10^n$ ; in scientific notation $1 \le a < 10$
Quotient: The result of a division, for example $4 \times 10^2$ in $(1.2 \times 10^5) \div (3 \times 10^2)$
Quotient rule: When dividing powers with the same base, subtract the exponents: $10^m \div 10^n = 10^{m-n}$
Normalize: Rewrite a result so the coefficient is between 1 and 10 without changing its value, e.g. $0.4 \times 10^3 = 4 \times 10^2$
Significant figures: The digits of a number that carry meaning about its precision; $2.33$ has 3 significant figures

Formula Card

Dividing in scientific notation

\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}

Divide the coefficients, subtract the exponents (top minus bottom).

Quotient rule

10^m \div 10^n = 10^{m-n}

Works for negative exponents too: $10^{-4} \div 10^{-7} = 10^{-4+7} = 10^3$.

Normalizing a small coefficient

c \times 10^k = (10c) \times 10^{k-1}

Use it when the coefficient $c$ is below 1: $0.4 \times 10^3 = 4 \times 10^2$.

Today's challengeOne problem, three tries

Dividing in Scientific Notation

Definition

Worked Examples

Common Mistakes

Interactive Visual

Divide, subtract the exponents, then normalize 0.4 × 10³

A quotient that never ends: round to 3 significant figures

Negative exponents: subtracting a negative adds

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Astronomy: Travel Time of Light

Geography: Population Density

Economics: Per-Person Share

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Dividing in scientific notation

Quotient rule

Normalizing a small coefficient

More in This Topic

Scientific Notation

Writing Numbers in Scientific Notation

Converting from Scientific Notation

Comparing Numbers in Scientific Notation