In Exercises 85–116, simplify each exponential expression.(20a⁻³b⁴c⁵/-2a⁻⁵b⁻²c)⁻²

Verified step by step guidance

1. Step 1: Simplify the expression inside the parentheses. Start by simplifying the coefficients: \( \frac{20}{-2} = -10 \).
2. Step 2: Apply the properties of exponents to simplify the variables. For \( a \), use \( a^{-3} / a^{-5} = a^{-3 - (-5)} = a^{2} \).
3. Step 3: Simplify the \( b \) terms: \( b^{4} / b^{-2} = b^{4 - (-2)} = b^{6} \).
4. Step 4: Simplify the \( c \) terms: \( c^{5} / c^{1} = c^{5 - 1} = c^{4} \).
5. Step 5: Apply the negative exponent to the entire expression: \( (-10a^{2}b^{6}c^{4})^{-2} = \frac{1}{(-10a^{2}b^{6}c^{4})^{2}} \).

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 quotient of powers (a^m / a^n = a^(m-n)), and the power of a power ( (a^m)^n = a^(m*n)). Understanding these rules is essential for simplifying expressions with exponents effectively.

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.

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 factoring polynomials and applying the rules of exponents. Mastery of this concept is vital for effectively simplifying complex expressions, especially those involving variables and exponents.