In Exercises 21–38, rewrite each expression with rational exponents.___⁵√11x

Verified step by step guidance

1. Identify the expression given: \( \sqrt[5]{11x} \).
2. Recall that a radical expression \( \sqrt[n]{a} \) can be rewritten using rational exponents as \( a^{1/n} \).
3. Apply this rule to the expression \( \sqrt[5]{11x} \), which becomes \( (11x)^{1/5} \).
4. Recognize that the expression \( (11x)^{1/5} \) is now written with a rational exponent.
5. Understand that this transformation allows for easier manipulation in algebraic operations, such as differentiation or integration.

Key Concepts

Rational Exponents

Rational exponents are a way to express roots using fractional powers. For example, the expression a^(m/n) represents the n-th root of a raised to the m-th power. This concept allows for a more compact representation of roots and is essential for rewriting expressions involving roots in algebra.

Radical Notation

Radical notation involves the use of the radical symbol (√) to denote roots of numbers. For instance, the expression √a represents the square root of a, while higher roots, such as cube roots, are denoted as ∛a. Understanding how to convert between radical and exponent notation is crucial for manipulating expressions in algebra.

Properties of Exponents

The properties of exponents are rules that govern how to manipulate expressions involving powers. Key properties include the product of powers, quotient of powers, and power of a power. These rules are vital for simplifying expressions and solving equations that involve exponents and roots.