In Exercises 117–124, simplify each exponential expression.(-4x³y⁻⁵)⁻²(2x⁻⁸y⁻⁵)

Verified step by step guidance

1. Step 1: Apply the power of a power property to the expression \((-4x^3y^{-5})^{-2}\). This means raising each factor inside the parentheses to the power of -2.
2. Step 2: Simplify \((-4)^{-2}\), \((x^3)^{-2}\), and \((y^{-5})^{-2}\) separately. Remember that \((a^m)^n = a^{m \cdot n}\).
3. Step 3: Combine the results from Step 2 to rewrite \((-4x^3y^{-5})^{-2}\) as a single expression.
4. Step 4: Simplify the expression \((2x^{-8}y^{-5})\) by recognizing that any negative exponent \(a^{-n}\) can be rewritten as \(\frac{1}{a^n}\).
5. Step 5: Multiply the simplified expressions from Step 3 and Step 4 together, combining like terms and applying the properties of exponents to simplify the final expression.

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, a^(-n) can be rewritten as 1/(a^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.

Combining Like Terms

Combining like terms involves simplifying expressions by merging terms that have the same variable raised to the same power. This process is essential in algebra to reduce expressions to their simplest form. In the context of exponential expressions, it helps in consolidating terms after applying the rules of exponents, leading to a clearer and more concise result.