Exponents and Indices

**Powers of 2.** A grain of rice is placed on the first square of a chessboard. The number of grains doubles on each successive square. (a) How many grains are on square 1, 2, 3, 4, 5? (b) Write a formula for the number of grains on square $n$. (c) How many grains are on square 32, in standard form (3 s.f.)?

**Cell biology.** A red blood cell has a diameter of about $7.5 \times 10^{-6}$ m. A human capillary has a diameter of about $9 \times 10^{-6}$ m. (a) How many times wider is the capillary than the red blood cell? Give your answer to 2 s.f. (b) Can the red blood cell fit through the capillary? Justify. (c) If a person has approximately $5 \times 10^{12}$ red blood cells, calculate the total volume in m³, treating each cell as a sphere with the diameter above. Use $V = \frac{4}{3}\pi r^3$ and give the answer in standard form (2 s.f.).

**Index laws investigation.** (a) Show that $\dfrac{x^5}{x^5} = 1$ and use this to explain why $x^0 = 1$. (b) Show that $\dfrac{x^3}{x^5} = \dfrac{1}{x^2}$ and use this to explain why $x^{-2} = \dfrac{1}{x^2}$. (c) Use the rule $(x^{1/2})^2 = x^1$ to explain why $x^{1/2} = \sqrt{x}$.

**Population growth.** A town's population in 2020 is $4.5 \times 10^4$. It grows at 3% per year. (a) Write the population $P_n$ after $n$ years in the form $P_n = a \cdot b^n$. (b) Predict the population in 2030 (in standard form to 3 s.f.). (c) After how many years does the population first exceed $7 \times 10^4$?

**Comparing magnitudes.** Place these in order from smallest to largest: $3 \times 10^{-2}, \quad 0.4, \quad 1.5 \times 10^{-1}, \quad \dfrac{1}{50}, \quad 6 \times 10^{-3}$

**Exponential decay.** A radioactive substance has a half-life of 5 days — every 5 days, half of it decays. A sample initially has mass 80 grams. (a) How much remains after 5, 10, 15, 20 days? (b) Write the mass $M$ after $5n$ days as a power expression. (c) After how many days is less than 1 gram remaining?

**Surd/fractional index practice.** (a) Write $\sqrt{x^6}$ using indices and simplify. (b) Write $\sqrt[3]{x^9}$ in simplest index form. (c) Simplify $\dfrac{\sqrt{x^5}}{x^{1/2}}$. (d) Hence solve $\sqrt{x} = x^{1/2} = 9$.

**The very large and very small.** A grain of sand has a typical mass of $4 \times 10^{-5}$ kg. The Earth has a mass of approximately $6 \times 10^{24}$ kg. (a) How many grains of sand would it take to equal the mass of the Earth? (b) The diameter of an atom is approximately $1 \times 10^{-10}$ m. The diameter of the Earth is approximately $1.3 \times 10^7$ m. How many atoms would span the Earth's diameter?

**Common errors.** A student writes the following. Find and correct any errors. (a) $(x^3)^2 = x^5$ (b) $x^3 + x^3 = x^6$ (c) $x^0 = 0$ (d) $(2x)^3 = 2x^3$

**Simplify and evaluate.** Let $a = 2$ and $b = 3$. (a) Calculate $a^{-1} + b^{-1}$, giving your answer as a fraction. (b) Calculate $(a + b)^{-1}$. (c) Comment on whether $(a + b)^{-1} = a^{-1} + b^{-1}$ in general.

**Index equation with manipulation.** Solve for $x$: $$2^{x+3} = 4^{x-1}.$$

**Light-years.** A light-year is the distance light travels in one year. Light travels at $3 \times 10^8$ m/s. (a) Show that 1 light-year is approximately $9.46 \times 10^{15}$ m (use 1 year = $3.154 \times 10^7$ s). (b) The nearest star (Proxima Centauri) is approximately 4.25 light-years away. How far is this in km, in standard form (3 s.f.)? (c) A spacecraft travels at $30000$ m/s. How many years would it take to reach Proxima Centauri?