Integer Exponents, Roots and Standard Form：練習問題

Evaluate the value of \((-3)^4\).

Solve the exponential equation \(9^{x-1} = 27^{x+2}\) for \(x\).

If \(5^x = 4\) and \(2^y = 125\), find the value of \(xy\).

Find the value of \(\sqrt{144} - \sqrt[3]{-64}\).

Simplify \(\dfrac{\left(3x^{-2}y^4\right)^{-2}}{9x^5 y^{-6}}\) and express your answer with positive indices.

Find the value of \(n\) that satisfies the equation:
\(\sqrt{3^n \cdot \sqrt{3^n \cdot \sqrt{3^n}}} = 27\sqrt{3}\)

Find the value of \(x\) that satisfies the following equation:
\(\sqrt[3]{8^x \cdot 4^{x+1}} = 16\)

Consider the following numerical expressions involving powers and square roots.

(a) Evaluate the expression \( 3^4 - 2^5 \).
(b) Find the value of \( \sqrt{144} + \sqrt{25} \).
(c) Using your answers from (a) and (b), determine the value of \( (3^4 - 2^5) \times (\sqrt{144} + \sqrt{25}) \).