In Exercises 21–38, rewrite each expression with rational exponents.__2x ³√y²

Verified step by step guidance

1. Identify the parts of the expression: \(2x\) and \(\sqrt[3]{y^2}\).
2. Recall that a radical expression \(\sqrt[n]{a^m}\) can be rewritten using rational exponents as \(a^{m/n}\).
3. Apply this rule to \(\sqrt[3]{y^2}\), rewriting it as \(y^{2/3}\).
4. Combine the rewritten expression with the remaining part of the original expression, \(2x\).
5. The expression with rational exponents is \(2x \cdot y^{2/3}\).

Key Concepts

Rational Exponents

Rational exponents are a way to express roots using fractional powers. For example, the expression a^(1/n) represents the n-th root of a. When dealing with rational exponents, the numerator indicates the power, while the denominator indicates the root. This concept is essential for rewriting expressions involving roots in a more algebraically manageable form.

Properties of Exponents

The properties of exponents are rules that govern how to manipulate expressions involving powers. Key properties include the product of powers (a^m * a^n = a^(m+n)), the power of a power ( (a^m)^n = a^(m*n)), and the power of a product ( (ab)^n = a^n * b^n). Understanding these properties is crucial for simplifying expressions and rewriting them with rational exponents.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots, and are often represented using the radical symbol (√). These expressions can be rewritten using rational exponents, which allows for easier manipulation in algebraic operations. Recognizing how to convert between radical and exponent forms is vital for solving problems that involve roots.