This checkpoint assesses your understanding of scientific notation (reading and writing it), operations and applications with very large and very small numbers, rounding to significant figures, fractional indices, and synthesis of all the index laws. It covers Lessons 15–20.
❓ Multiple Choice (10 questions)
1. Write 3.2 × 10⁵ as an ordinary number.
2. Which number is correctly written in scientific notation?
3. Write 93,000,000 in scientific notation.
4. Write 0.00045 in scientific notation.
5. Calculate (3 × 10⁴) × (2 × 10⁵), giving the answer in scientific notation.
6. Calculate (8 × 10⁶) ÷ (2 × 10²), giving the answer in scientific notation.
7. Round 2.4763 × 10⁸ to 3 significant figures.
8. Evaluate 27^(2/3) (a fractional index).
9. Calculate (5 × 10³) × (4 × 10²) and write the answer correctly in scientific notation.
10. Simplify (6a⁵b⁻²) ÷ (2a²b⁻⁵), giving your answer with positive indices only.
✍ Short Answer (4 questions)
11. Convert between ordinary numbers and scientific notation.
(a) Write 4,500,000 in scientific notation. (1 mark)
(b) Write 0.0007 in scientific notation. (1 mark)
(c) Write 6.02 × 10⁵ as an ordinary number. (1 mark)
(d) Write 3.8 × 10⁻³ as an ordinary number. (1 mark)
(e) Order these from smallest to largest: 2 × 10⁴, 5 × 10³, 9 × 10⁴. (2 marks)6 MARKS
12. Carry out each operation, giving every answer correctly in scientific notation.
(a) (4 × 10⁵) × (2 × 10³) (2 marks)
(b) (9 × 10⁸) ÷ (3 × 10²) (2 marks)
(c) (5 × 10⁴) × (6 × 10³) (2 marks)
(d) Explain why your answer to (c) must be rewritten so the first number is between 1 and 10. (1 mark)7 MARKS
13. Apply scientific notation to real measurements.
(a) Light travels at 3 × 10⁸ metres per second. How far does it travel in 60 seconds? Give your answer in scientific notation. (2 marks)
(b) A red blood cell is about 7 × 10⁻⁶ m wide. How many cells fit across a gap of 2.1 × 10⁻³ m? (2 marks)
(c) One water molecule has a mass of about 3.0 × 10⁻²³ g. Find the mass of 5 × 10²² molecules, giving your answer to 2 significant figures. (2 marks)
(d) Explain one advantage of scientific notation when working with quantities like these. (2 marks)8 MARKS
14. Fractional indices & synthesis:
(a) Evaluate 16^(1/2). (1 mark)
(b) Evaluate 8^(2/3). (2 marks)
(c) Write √x in index form. (1 mark)
(d) Simplify x^(1/2) × x^(1/3), giving your answer as a single power with a fractional index. (2 marks)
(e) Simplify (6a⁵b²) ÷ (2a²b⁻¹), giving your answer with positive indices only. (2 marks)8 MARKS
1. A3.2 × 10⁵ means move the decimal point 5 places right: 3.20000 → 320,000.
2. CScientific notation needs the first number to be between 1 and 10. Only 6.0 × 10⁴ qualifies (42, 12 are too big; 0.6 is too small).
3. B93,000,000 = 9.3 × 10⁷ (decimal moves 7 places; the number is large so the power is positive).
4. D0.00045 = 4.5 × 10⁻⁴ (decimal moves 4 places right to sit after the 4; small number → negative power).
5. BMultiply the front numbers (3 × 2 = 6) and add the powers (10⁴⁺⁵ = 10⁹): 6 × 10⁹.
6. ADivide the front numbers (8 ÷ 2 = 4) and subtract the powers (10⁶⁻² = 10⁴): 4 × 10⁴.
7. C3 significant figures keeps 2.47|63; the next digit is 6, so round up to 2.48 × 10⁸.
8. B27^(2/3) = (∛27)² = 3² = 9.
9. D5 × 4 = 20 and 10³⁺² = 10⁵, giving 20 × 10⁵. Since 20 is not between 1 and 10, rewrite as 2 × 10⁶.
10. C6 ÷ 2 = 3; a⁵⁻² = a³; b⁻²⁻⁽⁻⁵⁾ = b³, so the result is 3a³b³ (all indices positive).
Q11 (6 marks): (a) 4.5 × 10⁶ [1]. (b) 7 × 10⁻⁴ [1]. (c) 6.02 × 10⁵ = 602,000 [1]. (d) 3.8 × 10⁻³ = 0.0038 [1]. (e) Comparing powers and front numbers: 5 × 10³ (= 5,000), 2 × 10⁴ (= 20,000), 9 × 10⁴ (= 90,000). Order: 5 × 10³, 2 × 10⁴, 9 × 10⁴ [2].
Q12 (7 marks): (a) (4 × 2) × 10⁵⁺³ = 8 × 10⁸ [2]. (b) (9 ÷ 3) × 10⁸⁻² = 3 × 10⁶ [2]. (c) (5 × 6) × 10⁴⁺³ = 30 × 10⁷ = 3 × 10⁸ [2]. (d) The front number in scientific notation must be between 1 and 10; 30 is too large, so 30 × 10⁷ is rewritten as 3 × 10⁸ [1].
Q13 (8 marks): (a) (3 × 10⁸) × 60 = 180 × 10⁸ = 1.8 × 10¹⁰ m [2]. (b) (2.1 × 10⁻³) ÷ (7 × 10⁻⁶) = 0.3 × 10³ = 3 × 10² = 300 cells [2]. (c) (3.0 × 10⁻²³) × (5 × 10²²) = 15 × 10⁻¹ = 1.5 × 10⁰ = 1.5 g (2 s.f.) [2]. (d) Scientific notation lets you write very large and very small numbers compactly and compare their size at a glance using the power of 10, and it makes multiplying/dividing them easy (just operate on the front numbers and add/subtract the powers) [2].
Q14 (8 marks): (a) 16^(1/2) = √16 = 4 [1]. (b) 8^(2/3) = (∛8)² = 2² = 4 [2]. (c) √x = x^(1/2) [1]. (d) x^(1/2) × x^(1/3) = x^(1/2 + 1/3) = x^(3/6 + 2/6) = x^(5/6) [2]. (e) (6 ÷ 2)a⁵⁻²b²⁻⁽⁻¹⁾ = 3a³b³ [2].
Tick when you have finished all questions and checked your answers.