Indices：練習問題

Simplify the expression \(\frac{x^{-3} y^4}{x^2 y^0}\), where \(x \ne 0\) and \(y \ne 0\), expressing your answer using only positive indices.

Simplify the expression \( \frac{(2x^{-1}y^3)^3}{8x^5 y^{-2}} \), expressing your answer with positive indices only.

Given the following set of equations:
\( 2^a = 3 \)
\( 3^b = 4 \)
\( 4^c = 5 \)
\( 5^d = 6 \)
\( 6^e = 7 \)
\( 7^f = 8 \)
Find the value of the product \( abcdef \).

Simplify the expression \( \frac{k^{10}}{k^2} \).

Given that \(a^x = 5\), \(5^y = 7\) and \(7^z = a^3\), where \(a > 0\) and \(a \neq 1\), find the value of \(xyz\).

Evaluate the expression: \( \frac{3^5}{3^3} \times 3^0 \).

Simplify the expression \( \frac{(2x^{-3}y^2)^2}{8x^2y^{-5}} \), expressing your final answer using positive indices only.

Solve for the unknown variable \(x\) in the exponential equation: \( \frac{8^{2x-1} \cdot 4^{x+3}}{16^{x-2}} = 32^x \).

Consider the algebraic expression E given by:

$$ E = \frac{(9x^4 y^{-2} z)^2}{3x^{-5} y^3 z^0} $$

Assume that all variables are non-zero.

(a) Simplify the expression E, expressing your answer using only positive integral indices.

(b) Use your simplified expression from part (a) to evaluate E when \( x = 1 \) and \( y = 3 \).