Simplify each exponential expression in Exercises 23–64. $\frac{x^{30}}{x^{-10}}$

Verified step by step guidance

1. Identify the given expression: \(\frac{x^{30}}{x^{-10}}\).
2. Recall the quotient rule for exponents: when dividing like bases, subtract the exponents, i.e., \(\frac{a^m}{a^n} = a^{m-n}\).
3. Apply the quotient rule to the expression: \(x^{30 - (-10)}\).
4. Simplify the exponent subtraction: \$30 - (-10) = 30 + 10$.
5. Write the simplified expression as \(x^{40}\).

Key Concepts

Laws of Exponents

The laws of exponents provide rules for simplifying expressions involving powers. For division, the rule states that when dividing like bases, subtract the exponents: a^m / a^n = a^(m-n). This is essential for simplifying expressions such as x^30 / x^(-10).

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the positive exponent: a^(-n) = 1 / a^n. Understanding this helps in rewriting expressions with negative exponents into more manageable forms, which is crucial when simplifying x^30 / x^(-10).

Simplification of Algebraic Expressions

Simplification involves reducing expressions to their simplest form by applying algebraic rules. This includes combining like terms, applying exponent rules, and rewriting expressions for clarity. Mastery of simplification techniques ensures accurate and efficient problem solving.