In Exercises 21–38, rewrite each expression with rational exponents.__√x³

Verified step by step guidance

1. Recognize that the square root of a number can be expressed as a rational exponent. Specifically, \( \sqrt{x} \) is equivalent to \( x^{1/2} \).
2. Apply this understanding to the given expression \( \sqrt{x^3} \).
3. Rewrite the expression \( \sqrt{x^3} \) using rational exponents: \( (x^3)^{1/2} \).
4. Use the property of exponents that states \( (a^m)^n = a^{m \cdot n} \) to simplify the expression.
5. Multiply the exponents: \( x^{3 \cdot 1/2} = x^{3/2} \).

Key Concepts

Rational Exponents

Rational exponents are exponents that can be expressed as a fraction, where the numerator indicates the power and the denominator indicates the root. For example, x^(1/n) represents the n-th root of x. This concept allows us to rewrite expressions involving roots in a more manageable form, facilitating operations like multiplication and division.

Radical Notation

Radical notation is a way to express roots using the radical symbol (√). For instance, √x represents the square root of x. Understanding how to convert between radical notation and rational exponents is crucial for simplifying expressions and solving equations in algebra.

Properties of Exponents

The properties of exponents are rules that govern how to manipulate expressions with exponents. Key properties include the product of powers, power of a power, and the quotient of powers. These rules are essential for rewriting and simplifying expressions, especially when dealing with rational exponents and radicals.