In Exercises 107–114, simplify each exponential expression. Assume that variables represent nonzero real numbers. (x^−2 y)^−3/(x^2 y^−1)^3

Verified step by step guidance

1. Rewrite the expression using the property of exponents: \((a^m)^n = a^{m \cdot n}\).
2. Apply the property to the numerator: \((x^{-2} y)^{-3} = x^{(-2) \cdot (-3)} y^{-3}\).
3. Apply the property to the denominator: \((x^2 y^{-1})^3 = x^{2 \cdot 3} y^{-1 \cdot 3}\).
4. Simplify the exponents: \(x^6 y^{-3}\) in the numerator and \(x^6 y^{-3}\) in the denominator.
5. Cancel out the common terms in the numerator and denominator.

Key Concepts

Exponential Rules

Exponential rules are fundamental properties that govern the manipulation of expressions involving exponents. Key rules include the product of powers (a^m * a^n = a^(m+n)), the power of a power ( (a^m)^n = a^(m*n)), and the negative exponent rule (a^(-n) = 1/a^n). Understanding these rules is essential for simplifying expressions with exponents.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the opposite positive exponent. For example, x^(-n) can be rewritten as 1/x^n. This concept is crucial when simplifying expressions, as it allows for the transformation of negative exponents into a more manageable form, facilitating further simplification.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing fractions to their simplest form by canceling common factors in the numerator and denominator. This process often requires applying the rules of exponents and factoring. Mastery of this concept is vital for effectively simplifying complex expressions, especially those involving multiple variables and exponents.