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### (a) Simplify the following expressions:

**(i)** \[
\frac{ab^2c}{a^2b c^{-3}}
\]

We can simplify this by canceling terms and applying the laws of exponents.

- For \( a \): \( \frac{a}{a^2} = a^{1-2} = a^{-1} = \frac{1}{a} \)
- For \( b \): \( \frac{b^2}{b} = b^{2-1} = b \)
- For \( c \): \( \

Simplify the following expressions, solve the equations, and factorize the expressions as given in the provided image.

This guide provides step-by-step solutions for simplifying algebraic expressions, solving logarithmic equations, and factorizing quadratic expressions. Key concepts include exponent laws, logarithmic properties, and factorization techniques, suitable for Grades 9-11.

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