A Clean Division

Calculate $(8 \times 10^9) \div (2 \times 10^3)$.

Divide the coefficients: $8 \div 2 = 4$ = $4$

Subtract the exponents (quotient rule): $10^9 \div 10^3 = 10^{9-3} = 10^6$ = $10^6$

Combine and check the coefficient: $4 \times 10^6$, and $1 \le 4 < 10$ = Already normalized

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

The Coefficient Drops Below 1

Calculate $(1.2 \times 10^5) \div (3 \times 10^2)$ and write the answer in scientific notation.

Divide the coefficients: $1.2 \div 3 = 0.4$ = $0.4$

Subtract the exponents: $10^{5-2} = 10^3$ = $0.4 \times 10^3$

Check the coefficient: $0.4 < 1$, so this is not scientific notation yet = Normalize

Move the decimal point one place right, lower the exponent by 1: $0.4 \times 10^3 = (4 \times 10^{-1}) \times 10^3 = 4 \times 10^2$ = $4 \times 10^2$

Answer: $(1.2 \times 10^5) \div (3 \times 10^2) = 4 \times 10^2 = 400$

Dividing with Negative Exponents

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

Divide the coefficients: $6 \div 2 = 3$ = $3$

Write the exponent subtraction in full: $-4 - (-7)$ = Subtracting a negative

Subtracting a negative means adding: $-4 - (-7) = -4 + 7 = 3$ = $10^3$

Combine and check: $3 \times 10^3$, and $1 \le 3 < 10$ = $3 \times 10^3$

Answer: $(6 \times 10^{-4}) \div (2 \times 10^{-7}) = 3 \times 10^3 = 3000$

A Quotient That Never Ends

Calculate $(7 \times 10^8) \div (3 \times 10^3)$. Round the coefficient to 3 significant figures.

Divide the coefficients: $7 \div 3 = 2.333\ldots$ (the 3 repeats forever) = $2.333\ldots$

Round to 3 significant figures: Digits $2$, $3$, $3$; the next digit is $3$, so round down = $2.33$

Subtract the exponents: $10^{8-3} = 10^5$ = $10^5$

Combine, using the approximately equal sign: $2.33 \times 10^5$, and $1 \le 2.33 < 10$ = $\approx 2.33 \times 10^5$

Answer: $(7 \times 10^8) \div (3 \times 10^3) \approx 2.33 \times 10^5$

How Old Is Sunlight?

The Sun is about $1.5 \times 10^{11}$ m from Earth. Light travels $3 \times 10^8$ m per second. How many seconds does sunlight need to reach us?

Choose the operation: time = distance ÷ speed: $(1.5 \times 10^{11}) \div (3 \times 10^8)$ = Division

Divide the coefficients: $1.5 \div 3 = 0.5$ = $0.5$

Subtract the exponents: $10^{11-8} = 10^3$ = $0.5 \times 10^3$

Normalize: $0.5 \times 10^3 = 5 \times 10^2$ = $5 \times 10^2$ s

Make it easy to picture: $500 \div 60 \approx 8.3$ = About 8.3 minutes

Answer: $5 \times 10^2 = 500$ seconds, a little over 8 minutes

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

Why: 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).

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

Why: 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.

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

Why: 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 $400$.

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

Why: 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.

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.

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.

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.

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.

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.

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.

Q: Can the coefficient ever come out bigger than 10 when I divide?

A: 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.

Q: Why do we subtract the exponents?

A: 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 $10^m \div 10^n = 10^{m-n}$.

Q: What if both numbers have the same exponent?

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

Q: How many significant figures should I keep?

A: 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.