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.

· By the SOMATH team · 226 W 79th St, UWS · (646) 668-6151

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).

How to use this in a class (50–60 min):
  1. Warm-up: quick review of Class 9 exponent rules (10 min).
  2. Walk through the conversion rule and the four arithmetic patterns on the board with one example each (15 min).
  3. Students attempt the 25 questions with the answers hidden (25 min).
  4. Reveal answers, re-teach any that missed 3+ students, and end on an astronomy or biology application (10 min).

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 × 10ⁿ

A 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 asScientific notation?Why
3.7 × 10⁵Yes1 ≤ 3.7 < 10 and 5 is an integer
25 × 10³No25 is not between 1 and 10 — rewrite as 2.5 × 10⁴
0.6 × 10⁷No0.6 < 1 — rewrite as 6 × 10⁶
4.2 × 10⁻³Yes1 ≤ 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.
Example: 452,000
452,000. → move decimal 5 places left → 4.52
n = 5 (positive, because the original number is big)
→ 4.52 × 10⁵
Example: 0.0037
0.0037 → move decimal 3 places right → 3.7
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.
Example: 6.4 × 10⁶
Exponent is +6, so move the decimal 6 places right:
6.4 → 6,400,000
→ 6,400,000
Example: 8.2 × 10⁻⁴
Exponent is −4, so move the decimal 4 places left:
8.2 → 0.00082
→ 0.00082

5. Multiplying in scientific notation

Multiply coefficients, add exponents

(a × 10ᵐ) · (b × 10ⁿ) = (a · b) × 10ᵐ⁺ⁿ

This is just the product rule for exponents (from Class 9) applied to the powers of 10.

Example: (3 × 10⁴)(2 × 10⁵)
Coefficients: 3 · 2 = 6
Exponents: 4 + 5 = 9
→ 6 × 10⁹
Example that needs renormalizing: (5 × 10³)(4 × 10⁶)
Coefficients: 5 · 4 = 20  ← not between 1 and 10!
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⁻ⁿ

Quotient rule for exponents, applied to powers of 10.

Example: (8 × 10⁷) / (2 × 10³)
Coefficients: 8 / 2 = 4
Exponents: 7 − 3 = 4
→ 4 × 10⁴
Example that needs renormalizing: (3 × 10⁵) / (6 × 10²)
Coefficients: 3 / 6 = 0.5  ← not between 1 and 10!
Exponents: 5 − 2 = 3
Renormalize 0.5: 0.5 = 5.0 × 10⁻¹, so subtract 1 from the exponent.
→ 5.0 × 10²
Example with a negative exponent in the answer: (4 × 10²) / (8 × 10⁶)
Coefficients: 4 / 8 = 0.5
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:

  1. Rewrite one number so both have the same power of 10 (easiest: bump the smaller exponent up to match the larger).
  2. Add or subtract the coefficients.
  3. Renormalize so the coefficient is back between 1 and 10.
Example (exponents already match): 3.2 × 10⁶ + 4.5 × 10⁶
Coefficients: 3.2 + 4.5 = 7.7
Exponents: unchanged
→ 7.7 × 10⁶
Example (exponents differ): 5 × 10⁴ + 2 × 10³
Rewrite 2 × 10³ with exponent 4:
2 × 10³ = 0.2 × 10⁴
Now add: 5 × 10⁴ + 0.2 × 10⁴ = 5.2 × 10⁴
→ 5.2 × 10⁴
Example (subtraction that needs renormalizing): 1.02 × 10⁶ − 9.8 × 10⁵
Rewrite 9.8 × 10⁵ with exponent 6:
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 × 10ⁿᶜᴰ

Raise the coefficient to the power, and multiply the exponent on 10 by the power. Same power-of-a-power rule from Class 9.

Example: (2 × 10³)²
Coefficient: 2² = 4
Exponent: 3 · 2 = 6
→ 4 × 10⁶
Example that needs renormalizing: (3 × 10⁴)³
Coefficient: 3³ = 27  ← not between 1 and 10!
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.

Example: √(9 × 10⁶)
√9 = 3, and √(10⁶) = 10³ (halve the exponent).
→ 3 × 10³

9. Real-world applications

Where you actually see scientific notation

FieldQuantityValue
AstronomyEarth–Sun distance1.496 × 10⁸ km
AstronomyMass of the Sun1.989 × 10³⁰ kg
AstronomySpeed of light3 × 10⁸ m/s
BiologyDiameter of a bacterium~2 × 10⁻⁶ m
BiologyCells in the human body~3.7 × 10¹³
ChemistryAvogadro’s number6.022 × 10²³
ComputingBytes in 1 terabyte10¹²
ComputingTime 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#AnswerQ#AnswerQ#Answer
17.3 × 10⁴108 × 10⁸199 × 10⁸
24.6 × 10⁻⁴113 × 10¹²201.25 × 10¹¹
35.2 × 10⁹121 × 10⁴215 × 10² s (~500 s)
49.1 × 10⁻⁶133 × 10⁶221 × 10⁻³ m
5640,000143 × 10⁻⁴235 × 10⁵ (500,000)
60.00205158 × 10⁻⁹243.6×10⁸ bigger by 400×
7B and D166.8 × 10⁵25~3.3 × 10²⁵
88.4 × 10⁷173.5 × 10⁴
97.2 × 10⁻⁴186.7 × 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.

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