Corrigé
9.7 Indices and Scientific Notation
Pack A — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | Evaluate $2^5$. | 32 |
| 2 | Write $3^2 \times 3^5$ as a single power of $a$. | $3^7$ |
| 3 | Write $\dfrac{a^p}{a^q}$ as a single power. | $7^5$ |
| 4 | Write 32000 in standard form. | $3.2 \times 10^4$ |
| 5 | Write the decimal 0.004 in standard form. | $4 \times 10^{-3}$ |
| 6 | Evaluate $(7)^0$ and $(7)^{-2}$. | $1$ and $1/49$ |
| 7 | Write $(a^p)^q$ as a single power. | $2^{12}$ |
| 8 | Simplify $x^p \times x^q$. | $x^8$ |
| 9 | Write 0.5 × 10⁵ in standard form. | $5 \times 10^4$ |
| 10 | Evaluate $5^0$ and $5^1$. | 1 and 5 |
| 11 | Simplify $(a \times 10^p) \times (b \times 10^q)$. | $6 \times 10^9$ |
| 12 | Simplify $\dfrac{a \times 10^p}{b \times 10^q}$. | $4 \times 10^6$ |
| 13 | Simplify $\dfrac{x^p}{x^q}$. | $x^5$ |
| 14 | Evaluate $a^{1/2}$. | 5 |
| 15 | Write $35 \times 10^5$ in standard form. | $3.5 \times 10^6$ |
| 16 | Compute $5 \times 10^4 + 3 \times 10^3$ in standard form. | $5.3 \times 10^4$ |
| 17 | Evaluate $a^{-1}$. | $1/4 = 0.25$ |
| 18 | Simplify $(2x)^3$. | $8x^3$ |
| 19 | Convert: a light-year ≈ $9.5 \times 10^{12}$ km. Express in metres. | $9.5 \times 10^{15}$ m |
| 20 | Compute $(2 \times 10^3)^2$. | $4 \times 10^6$ |
| 21 | Simplify $\dfrac{(x^p)^q}{x^r}$. | $x^7$ |
| 22 | Write $35 \times 10^5$ in standard form. | $3.5 \times 10^6$ |
| 23 | Evaluate $a^{1/2}$ where a = 25. | 5 |
| 24 | UK population was $6.50 \times 10^7$ in 2016, growing 0.6% per year. Estimate the 2020 population (3 s.f.). | $6.66 \times 10^7$ |
| 25 | Compute $(3 \times 10^4) \times (4 \times 10^{-2})$ in standard form. | $1.2 \times 10^3$ |
| 26 | Convert $a \times 10^p$ to ordinary: $3.6 \times 10^{-3}$. | 0.0036 |
| 27 | Evaluate $a^{2/3}$ where a = 8. | 4 |
| 28 | Compute $\dfrac{6 \times 10^7}{2 \times 10^4}$ in standard form. | $3 \times 10^3$ |
| 29 | Simplify $(2x^3)^2 \times (3x)^2$. | $36 x^8$ |
| 30 | Order from smallest to largest: $3 \times 10^{-3}, 2 \times 10^{-4}, 4 \times 10^{-3}$. | $2 \times 10^{-4} < 3 \times 10^{-3} < 4 \times 10^{-3}$ |
| 31 | Solve $x^p = {rhs}$. | $x = 3$ |
| 32 | Evaluate $a^{-1/2}$. | $1/4$ |
| 33 | A nearby star is $4.2$ light-years away. 1 light-year ≈ $9.46 \times 10^{12}$ km. Find distance in km (3 s.f.). | $3.97 \times 10^{13}$ km |
| 34 | Solve $x^{1/3} = n$. | $x = 64$ |
| 35 | A bacteria culture starts at $2.5 \times 10^4$ cells, doubling every hour. (a) After 1 hour. (b) After 5 hours. (c) When does it exceed $10^8$? | (a) $5 \times 10^4$ (b) $8 \times 10^5$ (c) 12 hours |
| 36 | Evaluate $27^{-2/3}$. | $1/9$ |
| 37 | Solve $2^x = 32$. | $x = 5$ |
| 38 | Solve $x^4 = 81$ for real $x$. | $x = 3$ or $x = -3$ |
| 39 | Express $\sqrt[3]{27 a^9}$ in simplified index form. | $3 a^3$ |
| 40 | Compute $(10^3)^{1/3}$ and $(10^6)^{1/3}$. | $10$ and $10^2 = 100$ |
Pack B — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | Evaluate $3^4$. | 81 |
| 2 | Write $5^4 \times 5^3$ as a single power of $a$. | $5^7$ |
| 3 | Write $\dfrac{a^p}{a^q}$ as a single power. | $2^6$ |
| 4 | Write 7500000 in standard form. | $7.5 \times 10^6$ |
| 5 | Write the decimal 0.000 25 in standard form. | $2.5 \times 10^{-4}$ |
| 6 | Evaluate $(5)^0$ and $(5)^{-2}$. | $1$ and $1/25$ |
| 7 | Write $(a^p)^q$ as a single power. | $5^{12}$ |
| 8 | Simplify $x^p \times x^q$. | $x^9$ |
| 9 | Write 0.5 × 10⁵ in standard form. | $2.5 \times 10^{-2}$ |
| 10 | Evaluate $5^0$ and $5^1$. | 1 and 7 |
| 11 | Simplify $(a \times 10^p) \times (b \times 10^q)$. | $2 \times 10^{10}$ |
| 12 | Simplify $\dfrac{a \times 10^p}{b \times 10^q}$. | $3 \times 10^3$ |
| 13 | Simplify $\dfrac{x^p}{x^q}$. | $x^7$ |
| 14 | Evaluate $a^{1/2}$. | 9 |
| 15 | Write $35 \times 10^5$ in standard form. | $6 \times 10^3$ |
| 16 | Compute $5 \times 10^4 + 3 \times 10^3$ in standard form. | $2.4 \times 10^5$ |
| 17 | Evaluate $a^{-1}$. | $1/10 = 0.1$ |
| 18 | Simplify $(2x)^3$. | $9x^2$ |
| 19 | Convert: a light-year ≈ $9.5 \times 10^{12}$ km. Express in metres. | Same. |
| 20 | Compute $(2 \times 10^3)^2$. | $2.5 \times 10^9$ |
| 21 | Simplify $\dfrac{(x^p)^q}{x^r}$. | $x^7$ |
| 22 | Write $35 \times 10^5$ in standard form. | $6 \times 10^3$ |
| 23 | Evaluate $a^{1/2}$ where a = 25. | 10 |
| 24 | UK population was $6.50 \times 10^7$ in 2016, growing 0.6% per year. Estimate the 2020 population (3 s.f.). | $6.82 \times 10^7$ |
| 25 | Compute $(3 \times 10^4) \times (4 \times 10^{-2})$ in standard form. | $10^4 = 1 \times 10^4$ |
| 26 | Convert $a \times 10^p$ to ordinary: $3.6 \times 10^{-3}$. | 405 000 |
| 27 | Evaluate $a^{2/3}$ where a = 8. | 9 |
| 28 | Compute $\dfrac{6 \times 10^7}{2 \times 10^4}$ in standard form. | $2 \times 10^4$ |
| 29 | Simplify $(2x^3)^2 \times (3x)^2$. | Same. |
| 30 | Order from smallest to largest: $3 \times 10^{-3}, 2 \times 10^{-4}, 4 \times 10^{-3}$. | $3 \times 10^4 < 5 \times 10^4 < 2 \times 10^5$ |
| 31 | Solve $x^p = {rhs}$. | $x = \pm 2$ |
| 32 | Evaluate $a^{-1/2}$. | $1/10$ |
| 33 | A nearby star is $4.2$ light-years away. 1 light-year ≈ $9.46 \times 10^{12}$ km. Find distance in km (3 s.f.). | $8.14 \times 10^{13}$ km |
| 34 | Solve $x^{1/3} = n$. | $x = 125$ |
| 35 | A bacteria culture starts at $2.5 \times 10^4$ cells, doubling every hour. (a) After 1 hour. (b) After 5 hours. (c) When does it exceed $10^8$? | Same. |
| 36 | Evaluate $27^{-2/3}$. | $1/8$ |
| 37 | Solve $2^x = 32$. | $x = -3$ |
| 38 | Solve $x^4 = 81$ for real $x$. | $x = 2$ or $x = -2$ |
| 39 | Express $\sqrt[3]{27 a^9}$ in simplified index form. | $8 b^2$ |
| 40 | Compute $(10^3)^{1/3}$ and $(10^6)^{1/3}$. | Same. |
Problèmes — Solutions détaillées
**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) $x^5 \times x^{-3}$.
(a) $3^5$ (b) $x^5$ (c) $2^{12}$ (d) $x^2$
**UK population growth.** 2016: $6.50 \times 10^7$. Annual growth 0.6%. (a) Find the multiplier. (b) Estimate 2020 population (3 s.f.).
(a) 1.006 (b) $6.66 \times 10^7$
**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) $(2 \times 10^4)^3$
(a) $1.2 \times 10^5$ (b) $4 \times 10^5$ (c) $8 \times 10^{12}$
**Fractional indices.** Evaluate exactly: (a) $25^{1/2}$ (b) $8^{1/3}$ (c) $16^{3/4}$ (d) $27^{-2/3}$
(a) 5 (b) 2 (c) 8 (d) $1/9$
**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 > $10^8$?
(a) $5 \times 10^4$ (b) $8 \times 10^5$ (c) 12
**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.
(a) ≈ $1.97 \times 10^4$ s (b) ≈ 5.5 hours
**Solving index equations.** Solve each: (a) $2^x = 32$ (b) $5^x = 1/125$ (c) $x^4 = 81$ (all real solutions)
(a) $x = 5$ (b) $x = -3$ (c) $x = \pm 3$
**Large vs small.** Order from smallest to largest: $5 \times 10^4, 2 \times 10^5, 8 \times 10^3, 9 \times 10^4$.
$8 \times 10^3 < 5 \times 10^4 < 9 \times 10^4 < 2 \times 10^5$
**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 $5 \times 10^6$?
(a) 1.025 (b) ≈ $4.53 \times 10^6$ (c) Approximately 2029
**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.
(a) ≈ $5.99 \times 10^{23}$ (b) Same
**Algebraic indices.** Simplify each: (a) $(2x^3)^4$ (b) $\dfrac{12 x^7}{4 x^3}$ (c) $\sqrt[3]{27 x^6}$ (d) $(a^{1/2})^4$
(a) $16 x^{12}$ (b) $3 x^4$ (c) $3 x^2$ (d) $a^2$
**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.
(a) $2^9 = 512$ (b) $2^{31} \approx 2.15 \times 10^9$ (c) $2^{63} \approx 9.22 \times 10^{18}$