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Problem-solving Pack

Ecolint Campus des NationsMathematics
Year 9 · 9.7 Indices and Scientific Notation

Problem-solving Pack

Name: _________________________________
Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 12
Index laws round-up. Simplify each:

(a) 3×3×3×3×33 \times 3 \times 3 \times 3 \times 3 as a power of 3.
(b) x8x3\dfrac{x^8}{x^3}.
(c) (24)3(2^4)^3 as a power of 2.
(d) x5×x−3x^5 \times x^{-3}.

Working space

2Problem 2 of 12
UK population growth. 2016: 6.50×1076.50 \times 10^7. Annual growth 0.6%.

(a) Find the multiplier.
(b) Estimate 2020 population (3 s.f.).

Working space

3Problem 3 of 12
Standard form arithmetic. Work out each in standard form:

(a) (4×106)×(3×10−2)(4 \times 10^6) \times (3 \times 10^{-2})
(b) 6×1081.5×103\dfrac{6 \times 10^8}{1.5 \times 10^3}
(c) (2×104)3(2 \times 10^4)^3

Working space

4Problem 4 of 12
Fractional indices. Evaluate exactly:

(a) 251/225^{1/2}
(b) 81/38^{1/3}
(c) 163/416^{3/4}
(d) 27−2/327^{-2/3}

Working space

5Problem 5 of 12
Standard form: bacteria growth. Starts at 2.5×1042.5 \times 10^4. Doubles per hour.

(a) After 1 hour.
(b) After 5 hours (3 s.f.).
(c) After how many hours > 10810^8?

Working space

6Problem 6 of 12
Astronomical distances. Distance from Sun to Pluto ≈ 5.9×1095.9 \times 10^9 km. Distance light travels per second ≈ 3×1053 \times 10^5 km/s.

(a) Find the time for light to reach Pluto.
(b) Convert to hours.

Working space

7Problem 7 of 12
Solving index equations. Solve each:

(a) 2x=322^x = 32
(b) 5x=1/1255^x = 1/125
(c) x4=81x^4 = 81 (all real solutions)

Working space

8Problem 8 of 12
Large vs small. Order from smallest to largest: 5×104,2×105,8×103,9×1045 \times 10^4, 2 \times 10^5, 8 \times 10^3, 9 \times 10^4.

Working space

9Problem 9 of 12
Compound percentage with standard form. A population is 4.0×1064.0 \times 10^6 in 2020, growing 2.5% per year.

(a) Multiplier.
(b) Population in 2025 (3 s.f.).
(c) When does it exceed 5×1065 \times 10^6?

Working space

10Problem 10 of 12
Atomic mass. A hydrogen atom has mass ≈ 1.67×10−241.67 \times 10^{-24} g.

(a) How many hydrogen atoms in 1 g?
(b) Express in standard form.

Working space

11Problem 11 of 12
Algebraic indices. Simplify each:

(a) (2x3)4(2x^3)^4
(b) 12x74x3\dfrac{12 x^7}{4 x^3}
(c) 27x63\sqrt[3]{27 x^6}
(d) (a1/2)4(a^{1/2})^4

Working space

12Problem 12 of 12
Doubling chessboard. A famous problem: a chessboard has 64 squares. Place 1 grain on the first, 2 on the second, 4 on the third, ... doubling each time.

(a) How many grains on the 10th square?
(b) The 32nd square?
(c) The 64th square — give in standard form.

Working space