Problem-solving Pack

Mathematics

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 \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}

Working space

2Problem 2 of 12

UK population growth. 2016:

6.50 \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 \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

Working space

4Problem 4 of 12

Fractional indices. Evaluate exactly:

(a)

25^{1/2}

(b)

8^{1/3}

(c)

16^{3/4}

(d)

27^{-2/3}

Working space

5Problem 5 of 12

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

Working space

6Problem 6 of 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.

Working space

7Problem 7 of 12

Solving index equations. Solve each:

(a)

2^x = 32

(b)

5^x = 1/125

(c)

x^4 = 81

(all real solutions)

Working space

8Problem 8 of 12

Large vs small. Order from smallest to largest:

5 \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 \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

Working space

10Problem 10 of 12

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.

Working space

11Problem 11 of 12

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

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