A bicycle repair shop values a batch of spare parts for its annual stock check, ready for the accountant to review, using the expression \(\dfrac{(3x^2)^3}{...
Question 1 Report
A bicycle repair shop values a batch of spare parts for its annual stock check, ready for the accountant to review, using the expression \(\dfrac{(3x^2)^3}{x^{-1}}\), where \(x\) is a scaling factor used by the shop's stock system. Simplify \(\dfrac{(3x^2)^3}{x^{-1}}\) fully, giving your answer in the form \(kx^n\). (3)
Answer Details
Simplifying a mixed expression like this is done in stages: first apply the outer power to everything inside the brackets, then deal with dividing by a negative power.
Apply the power 3 to both the coefficient and the index inside the brackets: \((3x^2)^3 = 3^3\times x^{2\times3} = 27x^{6}\). [1 mark]
Dividing by \(x^{-1}\) is the same as multiplying by its reciprocal, \(x^{1}\), since \(\dfrac{1}{x^{-1}}=x\). [1 mark]
So \(\dfrac{27x^6}{x^{-1}} = 27x^{6}\times x = 27x^{6+1} = 27x^{7}\), giving \(k=27\) and \(n=7\). [1 mark]
Exam tip: dividing by a negative power always increases the index of \(x\) - if a final answer has a smaller index than the numerator started with, a sign error has crept in.
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