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Ecolint Campus des NationsMathématiques
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Problem-solving Pack

Ecolint Campus des NationsMathematics
Year 10 · Exponents and Indices

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
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 nn.
(c) How many grains are on square 32, in standard form (3 s.f.)?

Working space

2Problem 2 of 12
Cell biology. A red blood cell has a diameter of about 7.5×10−67.5 \times 10^{-6} m. A human capillary has a diameter of about 9×10−69 \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 5×10125 \times 10^{12} red blood cells, calculate the total volume in m³, treating each cell as a sphere with the diameter above. Use V=43πr3V = \frac{4}{3}\pi r^3 and give the answer in standard form (2 s.f.).

Working space

3Problem 3 of 12
Index laws investigation.

(a) Show that x5x5=1\dfrac{x^5}{x^5} = 1 and use this to explain why x0=1x^0 = 1.
(b) Show that x3x5=1x2\dfrac{x^3}{x^5} = \dfrac{1}{x^2} and use this to explain why x−2=1x2x^{-2} = \dfrac{1}{x^2}.
(c) Use the rule (x1/2)2=x1(x^{1/2})^2 = x^1 to explain why x1/2=xx^{1/2} = \sqrt{x}.

Working space

4Problem 4 of 12
Population growth. A town's population in 2020 is 4.5×1044.5 \times 10^4. It grows at 3% per year.

(a) Write the population PnP_n after nn years in the form Pn=a⋅bnP_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 7×1047 \times 10^4?

Working space

5Problem 5 of 12
Comparing magnitudes. Place these in order from smallest to largest:

3×10−2,0.4,1.5×10−1,150,6×10−33 \times 10^{-2}, \quad 0.4, \quad 1.5 \times 10^{-1}, \quad \dfrac{1}{50}, \quad 6 \times 10^{-3}

Working space

6Problem 6 of 12
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 MM after 5n5n days as a power expression.
(c) After how many days is less than 1 gram remaining?

Working space

7Problem 7 of 12
Surd/fractional index practice.

(a) Write x6\sqrt{x^6} using indices and simplify.
(b) Write x93\sqrt[3]{x^9} in simplest index form.
(c) Simplify x5x1/2\dfrac{\sqrt{x^5}}{x^{1/2}}.
(d) Hence solve x=x1/2=9\sqrt{x} = x^{1/2} = 9.

Working space

8Problem 8 of 12
The very large and very small. A grain of sand has a typical mass of 4×10−54 \times 10^{-5} kg. The Earth has a mass of approximately 6×10246 \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 1×10−101 \times 10^{-10} m. The diameter of the Earth is approximately 1.3×1071.3 \times 10^7 m. How many atoms would span the Earth's diameter?

Working space

9Problem 9 of 12
Common errors. A student writes the following. Find and correct any errors.

(a) (x3)2=x5(x^3)^2 = x^5
(b) x3+x3=x6x^3 + x^3 = x^6
(c) x0=0x^0 = 0
(d) (2x)3=2x3(2x)^3 = 2x^3

Working space

10Problem 10 of 12
Simplify and evaluate. Let a=2a = 2 and b=3b = 3.

(a) Calculate a−1+b−1a^{-1} + b^{-1}, giving your answer as a fraction.
(b) Calculate (a+b)−1(a + b)^{-1}.
(c) Comment on whether (a+b)−1=a−1+b−1(a + b)^{-1} = a^{-1} + b^{-1} in general.

Working space

11Problem 11 of 12
Index equation with manipulation. Solve for xx:

2x+3=4x−1.2^{x+3} = 4^{x-1}.

Working space

12Problem 12 of 12
Light-years. A light-year is the distance light travels in one year. Light travels at 3×1083 \times 10^8 m/s.

(a) Show that 1 light-year is approximately 9.46×10159.46 \times 10^{15} m (use 1 year = 3.154×1073.154 \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 3000030000 m/s. How many years would it take to reach Proxima Centauri?

Working space