Young Fermats · Algebra 1 Ignite · Class 10 · Scientific Notation & Applications · Grades 6–9 · NYC Math Class
Scientific Notation & Applications — 25 Practice Questions with Theory & Hidden Answers (Algebra 1 Ignite Class 10)
A complete class-ready lesson on scientific notation: converting to and from standard form, multiplying and dividing in scientific notation, adding and subtracting with matching and non-matching exponents, powers of a number in scientific notation, and real-world applications from astronomy, biology, and computing. Plus 25 practice questions with click-to-reveal step-by-step answers. Built for the SOMATH Algebra 1 Ignite class (Young Fermats, grades 6–9) on the Upper West Side of Manhattan.
This is Class 10 of the SOMATH Algebra 1 Ignite arc. In Class 9 we mastered the exponent rules — product, quotient, power of a power, zero, and negative exponents. Class 10 puts those rules to work: scientific notation is really just exponent rules applied to powers of 10, packaged as the way scientists, engineers, and calculators actually write very large and very small numbers.
Every question below has a hidden button that reveals the answer and the full reasoning, so a student can practice honestly and then check their thinking. Written by the same team that teaches Algebra 1 Ignite at SOMATH, a math-focused school on the Upper West Side of NYC run by cofounder Marcelo Ambrozio (Northwestern) and cofounder Vivianne Wright (Harvard).
- Warm-up: quick review of Class 9 exponent rules (10 min).
- Walk through the conversion rule and the four arithmetic patterns on the board with one example each (15 min).
- Students attempt the 25 questions with the answers hidden (25 min).
- Reveal answers, re-teach any that missed 3+ students, and end on an astronomy or biology application (10 min).
What’s in this lesson
- Why scientific notation exists
- The definition: a × 10ⁿ
- Converting standard form → scientific notation
- Converting scientific notation → standard form
- Multiplying in scientific notation
- Dividing in scientific notation
- Adding and subtracting in scientific notation
- Powers and roots in scientific notation
- Real-world applications
- 25 practice questions
- Answer key summary
- About SOMATH & Algebra 1 Ignite
1. Why scientific notation exists
The problem with really big and really small numbers
Try writing the mass of the Sun in kilograms: 1,989,000,000,000,000,000,000,000,000,000. Now try the mass of an electron: 0.000 000 000 000 000 000 000 000 000 911. Two problems:
- They’re easy to miscount — add or drop one zero and the answer is off by a factor of 10.
- They’re painful to compute with by hand.
Scientific notation solves both. The Sun’s mass becomes 1.989 × 10³⁰ kg. The electron becomes 9.11 × 10⁻³¹ kg. Now the size lives entirely in the exponent, and the meaningful digits sit up front.
2. The definition: a × 10ⁿ
Two strict rules
a × 10ⁿA number is in scientific notation when it satisfies both:
- 1 ≤ |a| < 10 — exactly one nonzero digit sits before the decimal point.
- n is an integer — positive, negative, or zero.
| Written as | Scientific notation? | Why |
|---|---|---|
| 3.7 × 10⁵ | Yes | 1 ≤ 3.7 < 10 and 5 is an integer |
| 25 × 10³ | No | 25 is not between 1 and 10 — rewrite as 2.5 × 10⁴ |
| 0.6 × 10⁷ | No | 0.6 < 1 — rewrite as 6 × 10⁶ |
| 4.2 × 10⁻³ | Yes | 1 ≤ 4.2 < 10 and −3 is an integer |
Renormalizing. If your coefficient a lands outside [1, 10) after a calculation, fix it by moving the decimal and adjusting the exponent:
- Coefficient too big (a ≥ 10)? Move the decimal left, add 1 to the exponent for each place.
- Coefficient too small (a < 1)? Move the decimal right, subtract 1 from the exponent for each place.
3. Converting standard form → scientific notation
Move the decimal, count the places
Slide the decimal point until exactly one nonzero digit sits to its left. The number of places you moved becomes the exponent n:
- Original number is ≥ 10? You moved the decimal left → n is positive.
- Original number is < 1? You moved the decimal right → n is negative.
- Original number is between 1 and 10? n = 0.
n = 5 (positive, because the original number is big)
→ 4.52 × 10⁵
n = −3 (negative, because the original number is small)
→ 3.7 × 10⁻³
Memory trick. Big number, big exponent. Small number, small (negative) exponent. If you ever get confused about the sign, ask yourself: is the original number huge, or tiny? That’s the sign of the exponent.
4. Converting scientific notation → standard form
Move the decimal, opposite direction
Going back is the reverse of going in. The exponent tells you how many places to move the decimal:
- Positive exponent n: move the decimal right n places (making the number bigger). Fill in zeros as needed.
- Negative exponent n: move the decimal left |n| places (making the number smaller). Fill in zeros in front as needed.
6.4 → 6,400,000
→ 6,400,000
8.2 → 0.00082
→ 0.00082
5. Multiplying in scientific notation
Multiply coefficients, add exponents
(a × 10ᵐ) · (b × 10ⁿ) = (a · b) × 10ᵐ⁺ⁿThis is just the product rule for exponents (from Class 9) applied to the powers of 10.
Exponents: 4 + 5 = 9
→ 6 × 10⁹
Exponents: 3 + 6 = 9
Renormalize 20: 20 = 2.0 × 10¹, so add 1 to the exponent.
→ 2.0 × 10¹⁰
6. Dividing in scientific notation
Divide coefficients, subtract exponents
(a × 10ᵐ) / (b × 10ⁿ) = (a / b) × 10ᵐ⁻ⁿQuotient rule for exponents, applied to powers of 10.
Exponents: 7 − 3 = 4
→ 4 × 10⁴
Exponents: 5 − 2 = 3
Renormalize 0.5: 0.5 = 5.0 × 10⁻¹, so subtract 1 from the exponent.
→ 5.0 × 10²
Exponents: 2 − 6 = −4
Renormalize 0.5 → 5.0 × 10⁻¹, so subtract 1: −4 − 1 = −5
→ 5.0 × 10⁻⁵
7. Adding and subtracting in scientific notation
Exponents must match first
Unlike multiplication and division, addition and subtraction do not have a shortcut. You must first make the exponents match, then add or subtract the coefficients:
- Rewrite one number so both have the same power of 10 (easiest: bump the smaller exponent up to match the larger).
- Add or subtract the coefficients.
- Renormalize so the coefficient is back between 1 and 10.
Exponents: unchanged
→ 7.7 × 10⁶
2 × 10³ = 0.2 × 10⁴
Now add: 5 × 10⁴ + 0.2 × 10⁴ = 5.2 × 10⁴
→ 5.2 × 10⁴
9.8 × 10⁵ = 0.98 × 10⁶
Subtract: 1.02 − 0.98 = 0.04
0.04 × 10⁶ is not normalized. 0.04 = 4.0 × 10⁻², so subtract 2 from the exponent: 6 − 2 = 4.
→ 4.0 × 10⁴
8. Powers and roots in scientific notation
Power of a power: multiply the exponent
(a × 10ⁿ)ᴰ = aᴰ × 10ⁿᶜᴰRaise the coefficient to the power, and multiply the exponent on 10 by the power. Same power-of-a-power rule from Class 9.
Exponent: 3 · 2 = 6
→ 4 × 10⁶
Exponent: 4 · 3 = 12
Renormalize 27 = 2.7 × 10¹, add 1 to the exponent: 12 + 1 = 13.
→ 2.7 × 10¹³
Square roots. The exponent on 10 must be even for a clean square-root answer. If it’s odd, rewrite the number first so the exponent becomes even.
→ 3 × 10³
9. Real-world applications
Where you actually see scientific notation
| Field | Quantity | Value |
|---|---|---|
| Astronomy | Earth–Sun distance | 1.496 × 10⁸ km |
| Astronomy | Mass of the Sun | 1.989 × 10³⁰ kg |
| Astronomy | Speed of light | 3 × 10⁸ m/s |
| Biology | Diameter of a bacterium | ~2 × 10⁻⁶ m |
| Biology | Cells in the human body | ~3.7 × 10¹³ |
| Chemistry | Avogadro’s number | 6.022 × 10²³ |
| Computing | Bytes in 1 terabyte | 10¹² |
| Computing | Time for one CPU cycle | ~3 × 10⁻¹⁰ s |
The application skill is reading a word problem, pulling out the numbers, converting to scientific notation, doing the arithmetic (usually multiplication or division), then converting back or interpreting the answer. Every question in the applications block below rehearses this move.
25 practice questions
Read the question. Try the problem in your head or on scratch paper. Then click the button to reveal the answer and the full reasoning.
Question 1
Write 73,000 in scientific notation.
Answer: 7.3 × 10⁴
Move the decimal 4 places left: 73,000. → 7.3. Original number is ≥ 10, so the exponent is positive: n = +4. → 7.3 × 10⁴.
Question 2
Write 0.00046 in scientific notation.
Answer: 4.6 × 10⁻⁴
Move the decimal 4 places right: 0.00046 → 4.6. Original number is < 1, so the exponent is negative: n = −4. → 4.6 × 10⁻⁴.
Question 3
Write 5,200,000,000 in scientific notation.
Answer: 5.2 × 10⁹
Move the decimal 9 places left: 5,200,000,000. → 5.2. Big number → positive exponent, n = 9. → 5.2 × 10⁹.
Question 4
Write 0.0000091 in scientific notation.
Answer: 9.1 × 10⁻⁶
Move the decimal 6 places right: 0.0000091 → 9.1. Small number → negative exponent, n = −6. → 9.1 × 10⁻⁶.
Question 5
Convert 6.4 × 10⁵ to standard form.
Answer: 640,000
Exponent is +5, so move the decimal 5 places right, filling in zeros: 6.4 → 640,000.
Question 6
Convert 2.05 × 10⁻³ to standard form.
Answer: 0.00205
Exponent is −3, so move the decimal 3 places left, filling in zeros in front: 2.05 → 0.00205.
Question 7
Which of these are written in correct scientific notation?
A) 15 × 10⁴
B) 6.7 × 10⁻²
C) 0.42 × 10⁵
D) 3.9 × 10⁷
Answer: B and D
Scientific notation requires 1 ≤ |a| < 10. A fails (15 is not < 10). C fails (0.42 is < 1). B and D both satisfy 1 ≤ a < 10 with integer exponents. Renormalized versions: A → 1.5 × 10⁵; C → 4.2 × 10⁴.
Question 8
Renormalize 84 × 10⁶ into proper scientific notation.
Answer: 8.4 × 10⁷
Coefficient 84 is too big. Move the decimal 1 place left: 84 → 8.4. That’s a factor of 10 removed, so add 1 to the exponent: 6 + 1 = 7. → 8.4 × 10⁷.
Question 9
Renormalize 0.72 × 10⁻³ into proper scientific notation.
Answer: 7.2 × 10⁻⁴
Coefficient 0.72 is too small. Move the decimal 1 place right: 0.72 → 7.2. That multiplied the coefficient by 10, so subtract 1 from the exponent: −3 − 1 = −4. → 7.2 × 10⁻⁴.
Question 10
Multiply: (4 × 10⁵)(2 × 10³). Give the answer in scientific notation.
Answer: 8 × 10⁸
Coefficients: 4 · 2 = 8. Exponents: 5 + 3 = 8. Coefficient is between 1 and 10, so no renormalizing needed. → 8 × 10⁸.
Question 11
Multiply: (6 × 10⁶)(5 × 10⁵). Give the answer in scientific notation.
Answer: 3 × 10¹²
Coefficients: 6 · 5 = 30. Exponents: 6 + 5 = 11. Coefficient 30 is too big, so renormalize: 30 = 3.0 × 10¹, add 1 to the exponent: 11 + 1 = 12. → 3 × 10¹².
Question 12
Multiply: (2.5 × 10⁻⁴)(4 × 10⁷). Give the answer in scientific notation.
Answer: 1 × 10⁴ (or equivalently, 10⁴)
Coefficients: 2.5 · 4 = 10. Exponents: −4 + 7 = 3. Renormalize 10 = 1.0 × 10¹, add 1 to the exponent: 3 + 1 = 4. → 1 × 10⁴.
Question 13
Divide: (9 × 10⁸) / (3 × 10²). Give the answer in scientific notation.
Answer: 3 × 10⁶
Coefficients: 9 / 3 = 3. Exponents: 8 − 2 = 6. → 3 × 10⁶.
Question 14
Divide: (1.5 × 10⁵) / (5 × 10⁸). Give the answer in scientific notation.
Answer: 3 × 10⁻⁴
Coefficients: 1.5 / 5 = 0.3. Exponents: 5 − 8 = −3. Renormalize 0.3 = 3.0 × 10⁻¹, subtract 1 from the exponent: −3 − 1 = −4. → 3 × 10⁻⁴.
Question 15
Divide: (6.4 × 10⁻²) / (8 × 10⁶). Give the answer in scientific notation.
Answer: 8 × 10⁻⁹
Coefficients: 6.4 / 8 = 0.8. Exponents: −2 − 6 = −8. Renormalize 0.8 = 8.0 × 10⁻¹, subtract 1 from the exponent: −8 − 1 = −9. → 8 × 10⁻⁹.
Question 16
Add: 4.7 × 10⁵ + 2.1 × 10⁵. Give the answer in scientific notation.
Answer: 6.8 × 10⁵
Exponents already match. Add the coefficients: 4.7 + 2.1 = 6.8. Keep the common power of 10. → 6.8 × 10⁵.
Question 17
Add: 3 × 10⁴ + 5 × 10³. Give the answer in scientific notation.
Answer: 3.5 × 10⁴
Exponents differ. Rewrite 5 × 10³ with exponent 4: 5 × 10³ = 0.5 × 10⁴. Now add: 3 + 0.5 = 3.5. → 3.5 × 10⁴.
Question 18
Subtract: 7.5 × 10⁶ − 8 × 10⁵. Give the answer in scientific notation.
Answer: 6.7 × 10⁶
Rewrite 8 × 10⁵ with exponent 6: 8 × 10⁵ = 0.8 × 10⁶. Subtract coefficients: 7.5 − 0.8 = 6.7. → 6.7 × 10⁶.
Question 19
Raise to a power: (3 × 10⁴)². Give the answer in scientific notation.
Answer: 9 × 10⁸
Coefficient: 3² = 9. Exponent: 4 · 2 = 8. Coefficient stays in [1, 10). → 9 × 10⁸.
Question 20
Raise to a power: (5 × 10³)³. Give the answer in scientific notation.
Answer: 1.25 × 10¹¹
Coefficient: 5³ = 125. Exponent: 3 · 3 = 9. Renormalize 125 = 1.25 × 10², add 2 to the exponent: 9 + 2 = 11. → 1.25 × 10¹¹.
Question 21
Astronomy application: Light travels at about 3 × 10⁸ meters per second. The Sun is about 1.5 × 10¹¹ meters from Earth. How many seconds does light take to reach Earth from the Sun?
Answer: 5 × 10² seconds — about 500 seconds, or 8 minutes 20 seconds.
Time = distance / speed = (1.5 × 10¹¹) / (3 × 10⁸).
Coefficients: 1.5 / 3 = 0.5. Exponents: 11 − 8 = 3.
Renormalize 0.5 → 5.0 × 10⁻¹, subtract 1 from the exponent: 3 − 1 = 2. → 5 × 10² seconds.
Interpretation: the sunlight hitting your face right now left the Sun about 8⅓ minutes ago.
Question 22
Biology application: A typical bacterium is about 2 × 10⁻⁶ meters across. If you lined up 500 of them end to end, how long would the line be in meters?
Answer: 1 × 10⁻³ meters (= 0.001 m = 1 mm).
First write 500 in scientific notation: 500 = 5 × 10².
Multiply: (5 × 10²)(2 × 10⁻⁶).
Coefficients: 5 · 2 = 10. Exponents: 2 + (−6) = −4.
Renormalize 10 = 1 × 10¹, add 1 to the exponent: −4 + 1 = −3. → 1 × 10⁻³ meters.
Interpretation: 500 bacteria in a row barely stretch across the tip of a pencil.
Question 23
Computing application: A hard drive stores 2 × 10¹² bytes. Each photo takes up about 4 × 10⁶ bytes. How many photos can the drive hold? Give the answer in scientific notation and in standard form.
Answer: 5 × 10⁵ photos (500,000).
Number of photos = total bytes / bytes per photo = (2 × 10¹²) / (4 × 10⁶).
Coefficients: 2 / 4 = 0.5. Exponents: 12 − 6 = 6.
Renormalize 0.5 = 5.0 × 10⁻¹, subtract 1 from the exponent: 6 − 1 = 5. → 5 × 10⁵ photos.
Standard form: 500,000 photos.
Question 24
Comparison challenge: Which is greater, and by how many times: 3.6 × 10⁸ or 9 × 10⁵?
Answer: 3.6 × 10⁸ is greater by a factor of 400 (or 4 × 10²).
Compare exponents first: 8 > 5, so 3.6 × 10⁸ is bigger by a factor of at least 10³ = 1000. But the coefficient of the smaller number (9) is larger than the coefficient of the bigger one (3.6), so the true ratio is smaller than 1000.
Ratio = (3.6 × 10⁸) / (9 × 10⁵).
Coefficients: 3.6 / 9 = 0.4. Exponents: 8 − 5 = 3.
Renormalize 0.4 = 4.0 × 10⁻¹, subtract 1: 3 − 1 = 2. → 4 × 10² = 400.
Sanity check: 400 · (9 × 10⁵) = 3600 · 10⁵ = 3.6 × 10⁸. ✅
Question 25
Modeling challenge: The mass of one water molecule is about 3 × 10⁻²⁶ kg. About how many water molecules are in 1 kilogram of water? Give the answer in scientific notation and briefly interpret the result.
Answer: ≈ 3.3 × 10²⁵ molecules.
Number of molecules = total mass / mass per molecule = 1 / (3 × 10⁻²⁶).
Rewrite 1 as 1 × 10⁰.
Divide: (1 / 3) × 10⁰⁻₀⁻₂⁶⁽ = 0.333… × 10²⁶.
Renormalize 0.333 = 3.33 × 10⁻¹, subtract 1: 26 − 1 = 25. → ≈ 3.3 × 10²⁵.
Interpretation: a single kilogram of water — about a quart — contains more molecules than there are stars in the observable universe (~10²²–10²&sup4;). Scientific notation is the only sensible way to write a number that big.
Answer key summary
| Q# | Answer | Q# | Answer | Q# | Answer |
|---|---|---|---|---|---|
| 1 | 7.3 × 10⁴ | 10 | 8 × 10⁸ | 19 | 9 × 10⁸ |
| 2 | 4.6 × 10⁻⁴ | 11 | 3 × 10¹² | 20 | 1.25 × 10¹¹ |
| 3 | 5.2 × 10⁹ | 12 | 1 × 10⁴ | 21 | 5 × 10² s (~500 s) |
| 4 | 9.1 × 10⁻⁶ | 13 | 3 × 10⁶ | 22 | 1 × 10⁻³ m |
| 5 | 640,000 | 14 | 3 × 10⁻⁴ | 23 | 5 × 10⁵ (500,000) |
| 6 | 0.00205 | 15 | 8 × 10⁻⁹ | 24 | 3.6×10⁸ bigger by 400× |
| 7 | B and D | 16 | 6.8 × 10⁵ | 25 | ~3.3 × 10²⁵ |
| 8 | 8.4 × 10⁷ | 17 | 3.5 × 10⁴ | ||
| 9 | 7.2 × 10⁻⁴ | 18 | 6.7 × 10⁶ |
About SOMATH & Algebra 1 Ignite
SOMATH — School of Math is a math-focused school on the Upper West Side of Manhattan for students in grades 1–12. Algebra 1 Ignite is the flagship Young Fermats track (grades 6–9), a 48-class arc that covers everything from linear equations through quadratics, functions, sequences, and statistics — the full Regents Algebra 1 and pre-SAT scope. This post is Class 10 of that arc.
Classes are taught by cofounder Marcelo Ambrozio (Northwestern-trained, 15+ years teaching math in NYC) and the SOMATH team.
Location: 226 W 79th St, 1st Floor, New York, NY 10024 (Upper West Side)
Phone: (646) 668-6151
Email: hello@schoolofmath.us
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