In Exercises 73–84, simplify each expression using the quotients-to-powers rule. (2/x)⁴

Verified step by step guidance

1. Identify the expression: \( \left( \frac{2}{x} \right)^4 \).
2. Apply the power of a quotient rule: \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \).
3. Raise the numerator to the power of 4: \( 2^4 \).
4. Raise the denominator to the power of 4: \( x^4 \).
5. Combine the results: \( \frac{2^4}{x^4} \).

Key Concepts

Quotients-to-Powers Rule

The quotients-to-powers rule states that when raising a fraction to a power, you can apply the exponent to both the numerator and the denominator separately. For example, (a/b)ⁿ = aⁿ/bⁿ. This rule simplifies calculations involving fractions raised to exponents, making it easier to manipulate algebraic expressions.

Exponent Rules

Exponent rules are a set of mathematical principles that govern how to handle powers and roots. Key rules include the product of powers, power of a power, and power of a product. Understanding these rules is essential for simplifying expressions involving exponents, as they provide a systematic way to combine and reduce terms.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form while maintaining equivalence. This process often includes combining like terms, applying exponent rules, and reducing fractions. Mastery of simplification is crucial in algebra, as it allows for clearer solutions and easier manipulation of equations.