Route planners write the ratio between the express and local bus fares for a shared journey as \(64^{\frac{2}{3}}\) hundred pounds per year, before rounding...
Question 1 Report
Route planners write the ratio between the express and local bus fares for a shared journey as \(64^{\frac{2}{3}}\) hundred pounds per year, before rounding the figure for the new public timetable that goes on display at every stop along the corridor.
Work out the value of \(64^{\frac{2}{3}}\). (3)
Answer Details
A fractional power \( a^{\frac{m}{n}} \) means "take the \(n\)th root of \(a\), then raise the result to the power \(m\)"; working with the root first usually keeps the numbers smaller.
The denominator of the fractional power, 3, means taking the cube root of 64 first: \( 64^{\frac{1}{3}} = 4 \), since \( 4^3 = 64 \) [1 mark]. The numerator, 2, means then squaring this result: \( 4^2 = 16 \) [1 mark]. Combining these two steps, \( 64^{\frac{2}{3}} = \left(64^{\frac{1}{3}}\right)^2 = 4^2 = 16 \) [1 mark].
Taking the cube root before squaring (rather than squaring 64 first and then trying to cube-root a much larger number) is the practical reason this order is always used when evaluating a fractional index by hand.
Download The App On Google Playstore
Everything you need to excel in your exams
