In Exercises 85–116, simplify each exponential expression.20x³/-5x⁴

Verified step by step guidance

1. Step 1: Identify the expression to simplify: \( \frac{20x^3}{-5x^4} \).
2. Step 2: Simplify the coefficients: Divide 20 by -5 to get -4.
3. Step 3: Simplify the exponents: Use the property \( \frac{x^a}{x^b} = x^{a-b} \) to simplify \( \frac{x^3}{x^4} \) to \( x^{3-4} = x^{-1} \).
4. Step 4: Combine the simplified coefficient and exponent: The expression becomes \( -4x^{-1} \).
5. Step 5: Rewrite the expression with positive exponents: \( -4x^{-1} \) can be rewritten as \( \frac{-4}{x} \).

Key Concepts

Exponential Expressions

Exponential expressions involve numbers raised to a power, indicating repeated multiplication. In algebra, these expressions can be simplified using the laws of exponents, which include rules for multiplying, dividing, and raising powers to powers. Understanding these rules is essential for simplifying expressions effectively.

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 recognizing equivalent expressions. Mastery of this concept is crucial for solving problems involving ratios of polynomials.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components, or factors, that can be multiplied to obtain the original polynomial. This is important in simplifying expressions, as it allows for the identification and cancellation of common factors. Techniques include finding the greatest common factor and using special product formulas.