Indices I：練習問題

Evaluate the expression:
\( 5^3 \times 2^3 \div 10^2 \)

Find the value of \( a \) that satisfies the equation:
\( 2^a = \sqrt{32} \times 2^{-3} \)

Find the value of the following numerical expression:
\( 16^{-\frac{1}{2}} \times 8^{\frac{2}{3}} \)

Determine the value of \(x\) that satisfies the following equation:
\(10^x = 0.001 \times 10^5\)

Find the value of \((3^2)^{-3} \times 3^4\) as a fraction in its simplest form.

Find the value of the following expression:
\( (0.25)^{-2} \)

Simplify the following expression and write your answer with positive indices:
\( \frac{(25x^4)^{-\frac{1}{2}}}{x^{-5}} \)

Solve for \( x \) in the following equation:
\( 5^x = \frac{1}{125} \)

Solve the following equations for \( x \):
(a) \( 2^x = 64 \)
(b) \( 5^{x-1} = \frac{1}{25} \)
(c) \( 10^x = 0.001 \)

Solve the equation:
\( \left( \frac{1}{4} \right)^{2x-3} = 8^{x+1} \)