A Positive Exponent

Write $4.5 \times 10^6$ as an ordinary number.

Read the exponent: $n = 6$ is positive, so the point moves $6$ places to the right = 6 places right

Use the digits you have: $4.5$ has one digit after the point, so the first hop uses the $5$: $45$ = 1 place used, 5 left

Fill in placeholder zeros: The other $5$ places become zeros: $45\,\underline{00000}$ = $4{,}500{,}000$

Check the size: $10^6$ is one million, and $4.5$ million is $4{,}500{,}000$ = Makes sense

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

A Negative Exponent

Write $3.2 \times 10^{-4}$ as an ordinary number.

Read the exponent: $n = -4$ is negative, so the point moves $4$ places to the left = 4 places left

Move the point one place at a time: $3.2 \to 0.32 \to 0.032 \to 0.0032 \to 0.00032$ = $0.00032$

Count the placeholder zeros: Between the point and the $3$ there are $4 - 1 = 3$ zeros = 3 zeros

Check the size: A negative exponent gives a positive number smaller than $1$ = $0.00032$ is between $0$ and $1$

Answer: $3.2 \times 10^{-4} = 0.00032$

Exponent Zero

Write $7.25 \times 10^0$ as an ordinary number.

Read the exponent: $n = 0$, and $10^0 = 1$ = Multiply by 1

Move the point zero places: $7.25 \times 1 = 7.25$ = $7.25$

Answer: $7.25 \times 10^0 = 7.25$: the number does not change

Distance from Earth to the Sun

The average distance from Earth to the Sun is $1.496 \times 10^8$ km. Write it as an ordinary number.

Read the exponent: $n = 8$, so move the point $8$ places right = 8 places right

Use the digits after the point: $1.496$ has $3$ digits after the point: $1496$ = 3 places used

Add the missing zeros: $8 - 3 = 5$ zeros: $1496\,\underline{00000}$ = $149{,}600{,}000$

Read it aloud: One hundred forty-nine million six hundred thousand = $149{,}600{,}000$ km

Answer: $1.496 \times 10^8$ km $= 149{,}600{,}000$ km

Reading a Calculator Display

A calculator shows 6.02E23. What number is that?

Translate the E: E23 means $\times 10^{23}$ = $6.02 \times 10^{23}$

Read the exponent: $n = 23$, move the point $23$ places right = 23 places right

Count the zeros: $6.02$ has $2$ digits after the point, so $23 - 2 = 21$ zeros follow $602$ = 21 zeros

Write it out: $602$ followed by $21$ zeros, $24$ digits in total = $602{,}000{,}000{,}000{,}000{,}000{,}000{,}000$

Answer: $6.02 \times 10^{23} = 602{,}000{,}000{,}000{,}000{,}000{,}000{,}000$ (Avogadro's number)

Mistake: Moving the point to the right for a negative exponent: $3.2 \times 10^{-4} = 32{,}000$

Why: A negative exponent means dividing by $10$ again and again, so the number gets smaller, not bigger.

Correct: Negative exponent, move left: $3.2 \times 10^{-4} = 0.00032$

Mistake: Adding $n$ zeros to the end: $4.5 \times 10^6 = 45{,}000{,}000$

Why: The exponent counts places the point moves, not zeros to write. The digit $5$ already uses one of the $6$ places.

Correct: Only $6 - 1 = 5$ zeros are needed: $4.5 \times 10^6 = 4{,}500{,}000$

Mistake: Writing $n$ zeros after the point: $3.2 \times 10^{-4} = 0.000032$

Why: The first hop to the left puts the point in front of the $3$ without adding a zero. That is one zero too many.

Correct: With a one-digit whole part, there are $|n| - 1$ zeros after the point: $0.00032$ (3 zeros)

Mistake: Thinking a negative exponent makes the number negative: $5 \times 10^{-3} = -5000$

Why: The sign of the exponent controls the size. Only a minus sign in front of the coefficient makes the number negative.

Correct: $5 \times 10^{-3} = 0.005$ (positive), while $-5 \times 10^{3} = -5000$

Astronomy

Astronomers write distances in scientific notation, but maps, books and news reports often give them in full.

Light travels about $3 \times 10^5$ km every second, which is $300{,}000$ km per second.

Biology and Medicine

Cells, bacteria and viruses are measured in tiny parts of a meter, so lab results often use negative exponents.

A bacterium about $2 \times 10^{-6}$ m long is $0.000002$ m, two millionths of a meter.

Calculators and Spreadsheets

When a result has too many digits for the screen, calculators and spreadsheets switch to E notation: 4.54E9 means $4.54 \times 10^9$.

A spreadsheet cell showing 2.5E-4 holds the number $0.00025$.

Q: How many zeros do I write for a positive exponent?

A: Subtract the number of digits after the point in the coefficient from the exponent. For $1.496 \times 10^8$: $8 - 3 = 5$ zeros, so $149{,}600{,}000$.

Q: How many zeros come after the point for a negative exponent?

A: When the coefficient is between $1$ and $10$, there are $|n| - 1$ zeros between the point and the first digit. For $7 \times 10^{-6}$: $5$ zeros, so $0.000007$.

Q: Can a negative exponent ever give a negative number?

A: No. $10^{-n}$ is always positive. A number is negative only if the coefficient has a minus sign, like $-2.5 \times 10^{-3} = -0.0025$.

Q: What does the E on my calculator mean?

A: E stands for "exponent" and means $\times 10^{\square}$. So 3.1E-5 is $3.1 \times 10^{-5} = 0.000031$. It has nothing to do with the number $e \approx 2.718$.