Numbers and their operations：練習問題

Express the decimal number \( 0.0000405 \) in standard form.

Given that \( 2^n \times 4^{n+1} = 32 \), find the value of \( n \).

Three bells toll at intervals of \( 12\text{ minutes} \), \( 15\text{ minutes} \), and \( 18\text{ minutes} \) respectively. If they all toll together at \( 09:00 \), at what time will they next toll together?

Find the Lowest Common Multiple (LCM) of \(24\) and \(60\).

Evaluate \( (4.5 \times 10^5) \times (2 \times 10^{-8}) \), giving your answer in standard form.

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

By using prime factorisation, find the smallest positive integer \( k \) such that \( 1260k \) is a perfect cube. Give your answer as an integer.

(a) Simplify \( (2a^3b^{-2})^4 \times \frac{1}{2}a^{-5}b \), leaving your answer in positive indices.
(b) The speed of light is approximately \( 3 \times 10^8 \) m/s. Calculate the time taken, in seconds, for light to travel a distance of \( 1.5 \times 10^{11} \) m, giving your answer in standard form.