A farmer's field is shown in the diagram, with the length and width labelled as powers of 10, in metres. length = \(10^3\) m\(10^2\) m© EAGLE BEACON GLOBAL ...
Question 1 Report
A farmer's field is shown in the diagram, with the length and width labelled as powers of 10, in metres.
- Work out the area of the field. Express your answer as a single power of 10. (2)
- State the numerical value of this area, in square metres. (1)
- The farmer converts the area to hectares by dividing by \(10^4\). Give the area in hectares as a single power of 10. (1)
Answer Details
This question multiplies two powers of 10 to find an area, then converts that area into hectares by a further division of powers of 10.
(a) Area is length times width, and multiplying powers with the same base adds the exponents: \[ \text{Area} = 10^3 \times 10^2 = 10^{3+2} = 10^5 \] [2 marks]
(b) Evaluating the power gives the numerical area: \[ 10^5 = 100000 \text{ m}^2 \] [1 mark]
(c) Dividing by \(10^4\) to convert to hectares subtracts the exponents: \[ 10^5 \div 10^4 = 10^{5-4} = 10^1 = 10 \text{ hectares} \] [1 mark]
Keeping the area as a power of 10 throughout makes the hectare conversion a one-step exponent subtraction, rather than a division of the large number 100000 by 10000.
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