In Exercises 85–116, simplify each exponential expression.6x²/2x⁻⁸

Verified step by step guidance

1. Identify the expression: \( \frac{6x^2}{2x^{-8}} \).
2. Simplify the coefficients: \( \frac{6}{2} = 3 \).
3. Apply the quotient rule for exponents: \( x^a / x^b = x^{a-b} \).
4. Subtract the exponents: \( x^{2 - (-8)} = x^{2 + 8} = x^{10} \).
5. Combine the simplified coefficient and the simplified variable expression: \( 3x^{10} \).

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.

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 that involve division of algebraic terms.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the opposite positive exponent (a^(-n) = 1/a^n). This concept is vital when simplifying expressions that contain negative exponents, as it allows for the transformation of the expression into a more manageable form. Recognizing and applying this rule is key to achieving the correct simplification.