Surds and exact calculation (Higher): แบบฝึกหัด

Calculate the value of \(5 + 3 \times 2^{2}\).

Find the value of \(n\) when \(32 \times 2^{n} = 2^{12}\).

The number of bacteria, \(N\), in a culture at time \(t\) hours is given by \(N = 500 \times 1.2^t\).
Calculate the percentage increase in the number of bacteria every 3 hours. Give your answer to 1 decimal place.

Which of the following numbers is a prime number?

Express 120 as a product of its prime factors. Write your answer in index notation.

Rationalise the denominator and simplify your answer fully:
\(\frac{10}{\sqrt{5} - 1}\)

The distance between Earth and the Sun is approximately \(1.5 \times 10^8\) km. Light travels at a speed of \(3.0 \times 10^5\) km/s. Calculate the time it takes for light to travel from the Sun to Earth, giving your answer in minutes and seconds.

Express the recurring decimal \(0.2\dot{3}\dot{6}\) as a fraction in its simplest form.

A solid metal cuboid has a mass of \(850 \text{ g}\) measured to the nearest \(10 \text{ g}\).
The dimensions of the cuboid are measured as:
Length = \(12.4 \text{ cm}\) (to 1 decimal place)
Width = \(8.5 \text{ cm}\) (to 1 decimal place)
Height = \(5.2 \text{ cm}\) (to 1 decimal place)

(a) Calculate the lower bound for the volume of the cuboid. Give your answer to 2 decimal places.
(b) Calculate the upper bound for the density of the metal. Give your answer to 3 significant figures.

Show that \( \frac{4 + \sqrt{3}}{2 - \sqrt{3}} \) can be written in the form \( a + b\sqrt{3} \), where \( a \) and \( b \) are integers.

Hence, or otherwise, solve the equation:
\( x(2 - \sqrt{3}) = 4 + \sqrt{3} \)