In Exercises 85–116, simplify each exponential expression. (-¼x⁻⁴y⁵z⁻¹)(-12x⁻³y⁻¹z⁴)

Verified step by step guidance

1. Identify the coefficients in the expression: -\frac{1}{4} and -12.
2. Multiply the coefficients: (-\frac{1}{4}) \times (-12).
3. Apply the product of powers property to the x terms: x^{-4} \times x^{-3}.
4. Apply the product of powers property to the y terms: y^{5} \times y^{-1}.
5. Apply the product of powers property to the z terms: z^{-1} \times z^{4}.

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 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, x^(-n) = 1/(x^n). This concept is crucial when simplifying expressions, as it allows for the conversion of negative exponents into a more manageable form, often leading to a clearer final expression.

Combining Like Terms

Combining like terms involves simplifying expressions by adding or subtracting terms that have the same variable raised to the same power. This process is vital in algebraic manipulation, as it helps to consolidate terms and reduce the complexity of the expression, making it easier to work with and understand.