In Exercises 21–38, rewrite each expression with rational exponents._∛5

Verified step by step guidance

1. Identify the given expression: \( \sqrt[3]{5} \).
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 given expression: \( \sqrt[3]{5} = 5^{1/3} \).
4. The expression \( 5^{1/3} \) is the equivalent form of \( \sqrt[3]{5} \) using rational exponents.
5. Verify that the expression is now in the form of a rational exponent.

Key Concepts

Rational Exponents

Rational exponents are a way to express roots using fractional powers. For example, the cube root of a number can be represented as that number raised to the power of one-third. This notation allows for easier manipulation of expressions, especially when combined with other algebraic operations.

Radical Notation

Radical notation involves using the radical symbol (√) to denote roots of numbers. The expression ∛5 represents the cube root of 5, which is the value that, when multiplied by itself three times, equals 5. Understanding how to convert between radical and exponent notation is essential for simplifying expressions.

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 crucial when rewriting expressions with rational exponents, as they help maintain the integrity of the mathematical relationships involved.