Mathematics • Year 10 • Unit 1 • Lesson 11
Scientific Notation, Skill Drill
Build fluency with the form a × 10ⁿ where 1 ≤ a < 10. Convert numbers from decimal form into scientific notation and back, and multiply or divide using the index laws. One step at a time, from a fully worked example through guided practice to independent problems.
1. I do, fully worked example
Read every step. Each one has a short reason on the right so you can see why, not just what.
Problem. Write 8,340,000 in scientific notation.
Step 1, Spot the form.
Scientific notation is a × 10ⁿ where 1 ≤ a < 10.
Reason: the mantissa must have exactly one non-zero digit before the decimal point.
Step 2, Place the decimal after the first non-zero digit.
8,340,000 → 8.340000 (decimal sits after the 8)
Reason: 8 is the first non-zero digit, so the new mantissa a = 8.34.
Step 3, Count how many places the decimal moved.
Moved 6 places to the LEFT → exponent = +6.
Reason: large numbers (≥ 10) get positive exponents.
Step 4, Write the answer.
8,340,000 = 8.34 × 10⁶
Reason: combine the new mantissa with the matching power of 10.
Answer: 8.34 × 10⁶
2. We do, fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank line. 4 marks
Problem. Write 0.00000725 in scientific notation.
Step 1, Spot the form: we need a × 10ⁿ with 1 ≤ a < 10. The first non-zero digit here is __________.
Step 2, Place the decimal after the first non-zero digit:
New mantissa a = ______________
Step 3, Count how many places the decimal moved. It moved __________ places to the __________ (left / right).
Step 4, Choose the sign of the exponent. Small numbers (less than 1) get a __________ (positive / negative) exponent.
Step 5, Write the final answer:
0.00000725 = ______________ × 10^(______)
3. You do, independent practice
Show your working in the space under each problem. The first four are foundation (one-step conversion). The middle two are standard (multiply/divide). The last two are extension (multi-step).
Foundation, single-step conversions
3.1 Write 45,600,000 in scientific notation. 1 mark
3.2 Write 0.00089 in scientific notation. 1 mark
3.3 Write 5.08 × 10⁻⁴ as an ordinary decimal. 1 mark
3.4 Write 2.6 × 10⁷ as an ordinary number (this is approximately Australia's population). 1 mark
Standard, multiply and divide
3.5 Calculate (4 × 10⁵) × (3 × 10³), giving your answer in scientific notation. 2 marks
3.6 Calculate (8 × 10⁹) ÷ (2 × 10⁵), giving your answer in scientific notation. 2 marks
Extension, push your thinking
3.7 A red blood cell has diameter 7 × 10⁻⁶ m. How many red blood cells, placed end-to-end, would stretch across a 1 mm gap? Give your answer in scientific notation. 3 marks
3.8 A student writes "45 × 10³ is in scientific notation". Their friend writes "0.45 × 10⁵ is in scientific notation". Both numbers equal 45,000. Which student is correct, and what is the correct scientific notation form of 45,000? Justify in one sentence. 2 marks
Section 2, We do (faded 0.00000725)
Step 1: first non-zero digit is 7.
Step 2: New mantissa a = 7.25.
Step 3: Decimal moved 6 places to the right.
Step 4: Small numbers (less than 1) get a negative exponent.
Step 5: 0.00000725 = 7.25 × 10⁻⁶.
3.1-45,600,000
Decimal moves 7 places left from after the final 0 to between 4 and 5: 4.56 × 10⁷.
3.2-0.00089
Decimal moves 4 places right to after the 8: 8.9 × 10⁻⁴.
3.3-5.08 × 10⁻⁴ as decimal
Move decimal 4 places left: 5.08 → 0.508 → 0.0508 → 0.00508 → 0.000508.
3.4-2.6 × 10⁷ as ordinary number
Move decimal 7 places right: 2.6 → 26 → 260 → 2,600 → 26,000 → 260,000 → 2,600,000 → 26,000,000.
3.5, (4 × 10⁵) × (3 × 10³)
Multiply mantissas: 4 × 3 = 12. Add exponents: 10⁵ × 10³ = 10⁸.
Intermediate: 12 × 10⁸. Adjust to proper form (mantissa must be < 10): 1.2 × 10⁹.
3.6, (8 × 10⁹) ÷ (2 × 10⁵)
Divide mantissas: 8 ÷ 2 = 4. Subtract exponents: 10⁹ ÷ 10⁵ = 10⁴.
Answer: 4 × 10⁴.
3.7, Red blood cells across 1 mm
1 mm = 1 × 10⁻³ m. Number of cells = (1 × 10⁻³) ÷ (7 × 10⁻⁶) = (1 ÷ 7) × 10⁻³⁻⁽⁻⁶⁾ = 0.1429 × 10³ ≈ 1.43 × 10² cells (about 143 cells).
Real anchor from the lesson: the diameter of a red blood cell is ≈ 7 × 10⁻⁶ m.
3.8, Which student is correct?
Neither is correct. In 45 × 10³ the mantissa 45 ≥ 10, and in 0.45 × 10⁵ the mantissa 0.45 < 1, both break the rule 1 ≤ a < 10. The correct scientific notation form of 45,000 is 4.5 × 10⁴.
Trap: the value can be right without the form being right.