In Exercises 85–116, simplify each exponential expression.x⁻⁷/x³

Verified step by step guidance

1. Step 1: Identify the expression to simplify, which is \( \frac{x^{-7}}{x^3} \).
2. Step 2: Apply the quotient rule for exponents, which states that \( \frac{a^m}{a^n} = a^{m-n} \).
3. Step 3: Substitute the exponents from the expression into the quotient rule: \( x^{-7-3} \).
4. Step 4: Simplify the exponent by performing the subtraction: \( x^{-10} \).
5. Step 5: Recognize that a negative exponent indicates a reciprocal, so \( x^{-10} = \frac{1}{x^{10}} \).

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 like x⁻⁷/x³.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the opposite positive exponent. For example, x⁻⁷ can be rewritten as 1/x⁷. This concept is crucial for simplifying expressions that contain negative exponents, allowing for easier manipulation and simplification of the overall expression.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing fractions to their simplest form by applying the rules of exponents and factoring. In the case of x⁻⁷/x³, this means combining the exponents in the numerator and denominator to achieve a single expression, which can then be expressed in a more manageable form, such as a positive exponent or a fraction.