結構與運算：練習題

Find the lowest common multiple (LCM) of \(2^3 \times 3^2 \times 5\) and \(2^2 \times 3^3 \times 7\).

If \(x = 12.4\) correct to 1 decimal place and \(y = 3.5\) correct to 1 decimal place, calculate the upper bound for the value of \(\frac{x}{y}\). Give your answer to 4 decimal places.

Calculate the value of the following expression, giving your answer as a decimal:
\(\frac{3}{4} \times (2^3 - 5) \div 0.5\)

Evaluate the following expression:
\((27^{\frac{2}{3}} + 16^{\frac{1}{4}})^{-1}\)

Simplify the following calculation and express your answer in standard form:
\(\frac{(4 \times 10^5)^2}{8 \times 10^{-3}}\)

Calculate the value of \((4 \times 10^{5}) \times (5 \times 10^{-2})\) and give your answer in standard form.

A rectangle has a length of 8.4 cm and a width of 5.2 cm, both rounded to 1 decimal place. Calculate the upper bound for the area of the rectangle.

In a class of 30 students, 20 like Mathematics (M), 15 like Art (A), and 2 like neither. Calculate \(n(M \cap A)\), the number of students who like both subjects.

The mass of the Earth is approximately \( 5.97 \times 10^{24} \) kg and the mass of the Moon is approximately \( 7.35 \times 10^{22} \) kg.
(a) Express the mass of the Earth as an ordinary number.
(b) Calculate how many times heavier the Earth is compared to the Moon. Give your answer to the nearest whole number.

Two numbers, \( X \) and \( Y \), are written as products of their prime factors:
\( X = 2^3 \times 3^a \times 5^2 \)
\( Y = 2^b \times 3^4 \times 7 \)
Given that the Highest Common Factor (HCF) of \( X \) and \( Y \) is 36, and the Lowest Common Multiple (LCM) is 126,000, find the values of \( a \) and \( b \).