數的運算、因數與倍數：練習題

Work out the value of \( 27^{\frac{2}{3}} \).

Simplify fully the expression \( (64a^{12}b^{-6})^{\frac{1}{3}} \).

A car travels a distance of \( 120 \text{ km} \), measured to the nearest \( 10 \text{ km} \). The time taken is \( 1.5 \text{ hours} \), measured to the nearest \( 0.1 \text{ hour} \). Calculate the lower bound for the average speed of the car, giving your answer to 2 decimal places.

Divide £\( 120 \) in the ratio \( 3 : 5 \). Find the value of the larger share.

The number of bacteria in a culture is given by the formula \(N = 500 \times 2^{t}\), where \(t\) is the time in hours. Calculate the time \(t\) when the number of bacteria reaches 16,000.

Simplify the expression \(\frac{10}{\sqrt{5}}\) by rationalising the denominator. Give your answer in the form \(a\sqrt{5}\).

Calculate the value of \( \left( \frac{64}{27} \right)^{-\frac{2}{3}} \times 2^2 \). Show your working clearly.

The distance between two stars is \( 8.64 \times 10^{13} \text{ km} \). A spacecraft travels at a constant speed of \( 4.8 \times 10^4 \text{ km/h} \). Calculate the time it would take for the spacecraft to travel between the stars. Give your answer in years, assuming \( 1 \text{ year} = 365 \text{ days} \). Give your answer in standard form to 3 significant figures.

A large water reservoir contains \(4.8 \times 10^7\) liters of water. During a drought, the water level decreases by \(1.2 \times 10^6\) liters every day for 15 days.

(a) Calculate the total amount of water lost over the 15 days, giving your answer in standard form.
(b) Calculate the volume of water remaining in the reservoir after 15 days. Give your answer in the form \(a \times 10^n\) where \(1 \le a < 10\) and \(n\) is an integer.

Three security lights are set to flash at different regular intervals. Light A flashes every \( 15 \) seconds, Light B flashes every \( 18 \) seconds, and Light C flashes every \( 24 \) seconds. At \( 20:00 \), all three lights flash at the same time.
(a) Find the next time that all three lights will flash at the same time.
(b) Calculate the total number of times all three lights will flash together between \( 20:00 \) and \( 21:00 \) inclusive.