A delivery company models the fuel efficiency of its vans using an expression involving indices, where \(a\) is the average load in tonnes and \(b\) is the ...
Question 1 Report
A delivery company models the fuel efficiency of its vans using an expression involving indices, where \(a\) is the average load in tonnes and \(b\) is the average speed in miles per hour.
- Simplify \(\left(\dfrac{a^4}{b^6}\right)^{\frac12}\). (2)
- Hence simplify \(\left(\dfrac{a^4}{b^6}\right)^{\frac12} \times \left(a^3 b^2\right)^{-1}\), giving your answer using positive indices only. (3)
Answer Details
A power of \(\frac12\) applied to a fraction takes the square root of the numerator and the denominator separately; combining that result with a further index expression then uses the usual laws for multiplying powers and for negative powers, converting any negative index to a positive one only at the end.
- \(\left(\dfrac{a^4}{b^6}\right)^{\frac12}=\dfrac{(a^4)^{1/2}}{(b^6)^{1/2}}=\dfrac{a^2}{b^3}\) [2 marks]
- \((a^3b^2)^{-1}=a^{-3}b^{-2}\), so multiplying: \(a^{2-3}b^{-3-2}\) [1 mark]; \(=a^{-1}b^{-5}\) [1 mark]; writing with positive indices, \(=\dfrac{1}{ab^5}\) [1 mark]
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