9.7 Indices and Scientific Notation

Show all working. Partial marks are given for method.

1

**Index laws round-up.** Simplify each: (a) $3 \times 3 \times 3 \times 3 \times 3$ as a power of 3. (b) $\dfrac{x^8}{x^3}$. (c) $(2^4)^3$ as a power of 2. (d) $x^5 \times x^{-3}$.

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2

**UK population growth.** 2016: $6.50 \times 10^7$. Annual growth 0.6%. (a) Find the multiplier. (b) Estimate 2020 population (3 s.f.).

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3

**Standard form arithmetic.** Work out each in standard form: (a) $(4 \times 10^6) \times (3 \times 10^{-2})$ (b) $\dfrac{6 \times 10^8}{1.5 \times 10^3}$ (c) $(2 \times 10^4)^3$

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4

**Fractional indices.** Evaluate exactly: (a) $25^{1/2}$ (b) $8^{1/3}$ (c) $16^{3/4}$ (d) $27^{-2/3}$

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5

**Standard form: bacteria growth.** Starts at $2.5 \times 10^4$. Doubles per hour. (a) After 1 hour. (b) After 5 hours (3 s.f.). (c) After how many hours > $10^8$?

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6

**Astronomical distances.** Distance from Sun to Pluto ≈ $5.9 \times 10^9$ km. Distance light travels per second ≈ $3 \times 10^5$ km/s. (a) Find the time for light to reach Pluto. (b) Convert to hours.

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7

**Solving index equations.** Solve each: (a) $2^x = 32$ (b) $5^x = 1/125$ (c) $x^4 = 81$ (all real solutions)

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8

**Large vs small.** Order from smallest to largest: $5 \times 10^4, 2 \times 10^5, 8 \times 10^3, 9 \times 10^4$.

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9

**Compound percentage with standard form.** A population is $4.0 \times 10^6$ in 2020, growing 2.5% per year. (a) Multiplier. (b) Population in 2025 (3 s.f.). (c) When does it exceed $5 \times 10^6$?

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10

**Atomic mass.** A hydrogen atom has mass ≈ $1.67 \times 10^{-24}$ g. (a) How many hydrogen atoms in 1 g? (b) Express in standard form.

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11

**Algebraic indices.** Simplify each: (a) $(2x^3)^4$ (b) $\dfrac{12 x^7}{4 x^3}$ (c) $\sqrt[3]{27 x^6}$ (d) $(a^{1/2})^4$

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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.

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