Simplify each expression, and eliminate any negative...

Simplify each expression, and eliminate any negative exponents.
$$
\text { (a) }\left(\frac{x^{4} z^{2}}{4 y^{5}}\right)\left(\frac{2 x^{3} y^{2}}{z^{3}}\right)^{2} \quad \text { (b) } \frac{\left(r s^{2}\right)^{3}}{\left(r^{-3} s^{2}\right)^{2}}
$$

Key Concepts

Negative Exponent Rule

The negative exponent rule states that a negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. This understanding is necessary for rewriting expressions without negative exponents and for simplifying expressions into standard form.

Power of a Product Rule

This rule explains that when a product of factors is raised to an exponent, each factor in the product is raised to that exponent individually. It is crucial for breaking down complex expressions into simpler parts that can be separately simplified.

Quotient of Powers Rule

When dividing two exponential expressions with the same base, the quotient of the powers rule is applied by subtracting the exponent in the denominator from the exponent in the numerator. This concept is important in reducing fractional expressions involving exponents.

Product of Powers Rule

This rule states that when multiplying two powers with the same base, you add the exponents. It is fundamental in simplifying expressions because it allows for the consolidation of terms and elimination of unnecessary complexity in algebraic expressions involving exponents.

Power of a Power Rule

The power of a power rule indicates that when an exponentiated term is raised to another exponent, the two exponents multiply. This rule is essential for simplifying nested exponents and is often used in situations where expressions are raised to additional powers.

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