In Exercises 117–124, simplify each exponential expression.(2x²y⁴)⁻¹(4xy³)⁻³ / (x²y)⁻⁵(x³y²)⁴

Verified step by step guidance

1. Step 1: Apply the negative exponent rule to each term. For example, \((a^m)^n = a^{m \cdot n}\) and \((a^m)^{-n} = \frac{1}{(a^m)^n}\).
2. Step 2: Simplify each part separately. For \((2x^2y^4)^{-1}\), it becomes \(\frac{1}{2x^2y^4}\). For \((4xy^3)^{-3}\), it becomes \(\frac{1}{(4xy^3)^3}\).
3. Step 3: Simplify the denominator. For \((x^2y)^{-5}\), it becomes \(x^{10}y^5\). For \((x^3y^2)^4\), it becomes \(x^{12}y^8\).
4. Step 4: Combine the expressions. Multiply the simplified numerators and divide by the simplified denominators.
5. Step 5: Use the properties of exponents to combine like terms. Remember \(a^m \cdot a^n = a^{m+n}\) and \(\frac{a^m}{a^n} = a^{m-n}\).

Key Concepts

Exponential Rules

Exponential rules govern how to manipulate 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 power of a product ( (ab)^n = a^n * b^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, a^(-n) = 1/(a^n). This concept is crucial when simplifying expressions, as it allows for the transformation of negative exponents into a more manageable form, often leading to a clearer final expression.

Combining Exponential Expressions

When simplifying expressions that involve multiple exponential terms, it is important to combine like bases and apply the rules of exponents systematically. This includes adding or subtracting exponents when multiplying or dividing terms with the same base. Mastery of this concept is key to achieving a simplified form of complex exponential expressions.