Indices, surds and standard form

Evaluate — 2 marks

A scientist is studying the mass of particles in a laboratory experiment. She measures the mass of a single carbon atom as 1.99 × 10⁻²⁶ kg. She needs to calculate the total mass of a sample containing 5 × 10²³ carbon atoms.

Evaluate (1.99 × 10⁻²⁶) × (5 × 10²³). Give your answer in standard form. [2 marks]

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Correctly multiplies the coefficients: 1.99 × 5 = 9.95 (or equivalent working shown)
Correctly adds the indices and gives final answer in standard form: 9.95 × 10⁻³ (or 9.95 × 10⁻³ kg)

Explain — 4 marks

A scientist is studying the spread of bacteria in a laboratory sample. The initial population is 2.5 × 10³ cells. After each hour, the population multiplies by a factor of 2². The scientist needs to express very large numbers and understand how the population grows.

(a) The bacteria population after 3 hours can be calculated using the expression: 2.5 × 10³ × (2²)³. Explain why the exponent in the bracket is cubed. [1 mark]
(b) Simplify 2.5 × 10³ × (2²)³ and express your answer in standard form. [2 marks]
(c) Explain why expressing the final answer in standard form (rather than as an ordinary number) is useful when dealing with very large populations in scientific research. [1 mark]

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(a) Because the population multiplies by 2² each hour, and this happens for 3 hours / the exponent represents the number of hours elapsed
(b) Correctly simplifies (2²)³ to 2⁶ (or calculates 2⁶ = 64)
(b) Correctly calculates 2.5 × 10³ × 64 = 1.6 × 10⁵ (or equivalent working showing 250,000 × 64 = 16,000,000 then converting to standard form)
(c) Standard form makes very large numbers easier to read / write / compare / use in calculations / shows the order of magnitude clearly

Show — 3 marks

A company installs solar panels on houses. The area of one solar panel is 2.5 × 10⁻¹ m². A house needs enough panels to cover a total area of 1.5 × 10¹ m².

(a) Show that the number of solar panels needed is 60. [2 marks]
(b) Each solar panel costs £3.2 × 10². Show that the total cost for 60 panels is £1.92 × 10⁴. [1 mark]

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(a) Method: 1.5 × 10¹ ÷ 2.5 × 10⁻¹ or equivalent
(a) Correct calculation leading to 60
(b) Correct multiplication: 60 × 3.2 × 10² = 192 × 10² = 1.92 × 10⁴

Explain — 2 marks

A mathematics student is revising index laws for an exam. They are practising simplifying expressions using the rules of indices instead of calculating large numbers directly.

(a) Simplify fully $5^8 \div 5^3$ [1 mark]
(b) Explain how you can use a rule of indices to find the answer to part 01.1 without working out the value of $5^8$ or $5^3$ [1 mark]

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(a) $5^5$ or 3125
(b) States that when dividing powers with the same base, subtract the indices/powers
(b) e.g. $8 - 3 = 5$ so $5^{8-3} = 5^5$

State — 4 marks

A computer server stores data in bytes. The capacity of one hard drive is $2^{40}$ bytes. A data centre contains $2^{10}$ identical hard drives.

(a) State $2^{40}$ in standard form. [2 marks]
(b) State the total storage capacity of all the hard drives in the data centre, giving your answer as a single power of 2. [2 marks]

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(a) $2^{40} = 1099511627776$ or equivalent calculation shown
(a) $1.099511627776 \times 10^{12}$ or $1.1 \times 10^{12}$ (accept $1.0995... \times 10^{12}$)
(b) Correct method: $2^{40} \times 2^{10}$ or $2^{40+10}$ seen
(b) $2^{50}$

Evaluate — 2 marks

Explain — 4 marks

Show — 3 marks

Explain — 2 marks

State — 4 marks

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