Simplify each exponential expression. Assume that variables represent nonzero real numbers. $\frac{(x^{-2} y)^{-3}}{(x^{2} y^{-1})^{3}}$

Verified step by step guidance

1. Start by rewriting the given expression clearly: \(\frac{(x^{-2} y)^{-3}}{(x^{2} y^{-1})^{3}}\).
2. Apply the power of a power rule, which states that \((a^{m})^{n} = a^{m \times n}\), to both the numerator and the denominator separately.
3. For the numerator: \((x^{-2} y)^{-3} = x^{-2 \times (-3)} y^{1 \times (-3)} = x^{6} y^{-3}\).
4. For the denominator: \((x^{2} y^{-1})^{3} = x^{2 \times 3} y^{-1 \times 3} = x^{6} y^{-3}\).
5. Rewrite the expression as \(\frac{x^{6} y^{-3}}{x^{6} y^{-3}}\) and then apply the quotient rule for exponents, which states \(\frac{a^{m}}{a^{n}} = a^{m-n}\), to simplify the expression.

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers. Key rules include multiplying exponents when raising a power to another power, adding exponents when multiplying like bases, and subtracting exponents when dividing like bases. Understanding these rules is essential for simplifying complex exponential expressions.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, x^(-n) equals 1/x^n. Recognizing and correctly applying this rule helps simplify expressions with negative powers by rewriting them as fractions.

Simplifying Rational Expressions with Exponents

When simplifying a fraction involving exponential expressions, apply exponent rules separately to numerator and denominator, then combine results. This often involves distributing exponents over products and quotients and reducing common factors. Mastery of this process is crucial for simplifying expressions like the given problem.