Standard form and laws of exponents：練習問題

Solve the exponential equation \( 9^x = 27^{x-1} \) for \( x \).

The first three terms of an arithmetic sequence are \( 2k+1, 3k, 5k-2 \). Find the value of \( k \).

The polynomial \( P(x) = x^3 - ax^2 + bx - 6 \) has roots \( \alpha, \beta, \gamma \). If \( \alpha + \beta + \gamma = 6 \) and \( \frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} = \frac{3}{2} \), find the value of \( a-b \).

Which of the following expressions is equivalent to \( 2\log_a(x) - \log_a(y) \) for \( x, y, a > 0 \)?

If \( 2^x = 3^y = 12^z \), express \( z \) in terms of \( x \) and \( y \).

Find the 20th term of an arithmetic sequence where the first term is 5 and the common difference is -3.

Determine the coefficient of the \( x^3y^2 \) term in the expansion of \( (2x + y)^5 \).

Find the modulus and argument of the complex number \( z = \frac{(1-i\sqrt{3})^3}{(2+2i)^2} \).

Find the coefficient of \( x^4 \) in the expansion of \( (2x + 1)^7 \).

Consider the expansion of the expression \( \left( 3x^2 - \frac{k}{x} \right)^6 \), where \( k \) is a positive constant.

a) Given that the constant term in this expansion is 1215, show that \( k = \sqrt{3} \).
b) Using the value of \( k \) found in part (a), determine the coefficient of the term in \( x^3 \).