Indices II：练习题

Simplify \( y^7 \times y^{-3} \).

Find the value of \( x \) such that \( \frac{1}{27} = 3^{x-2} \).

Solve the following index equation for \( x \):
\( \left(\frac{1}{9}\right)^{x+2} = 27^{x-1} \)

Solve for \( x \): \( 8^{x+2} = 16^{x-1} \).

Solve the following index equation for \( x \):
\( 5^{x+1} + 5^x = 150 \)

Calculate the exact value of \(3^{-3}\).

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

Solve for $$x$$: $$ 4^{2x-1} = 8^{x+2} $$.

(a) Solve the equation \( 3^{5x-2} = 81 \).

(b) Find the value of \( k \) such that \( (25^{k+1}) = 5 \times (\sqrt{5})^k \).