In Exercises 79–112, use rational exponents to simplify each expression. If rational exponents appear after simplifying, write the answer in radical notation. Assume that all variables represent positive numbers.___ __⁴√a²b ⋅ ³√ab

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1. Step 1: Express each radical using rational exponents. The expression \( \sqrt[4]{a^2b} \) can be written as \( (a^2b)^{1/4} \) and \( \sqrt[3]{ab} \) can be written as \( (ab)^{1/3} \).
2. Step 2: Apply the property of exponents \( (x^m \cdot x^n = x^{m+n}) \) to combine the expressions: \( (a^2b)^{1/4} \cdot (ab)^{1/3} = a^{2/4}b^{1/4} \cdot a^{1/3}b^{1/3} \).
3. Step 3: Simplify the exponents by adding them: \( a^{2/4 + 1/3} \cdot b^{1/4 + 1/3} \).
4. Step 4: Find a common denominator to add the exponents: \( a^{3/6 + 2/6} \cdot b^{3/12 + 4/12} \).
5. Step 5: Simplify the exponents: \( a^{5/6} \cdot b^{7/12} \). Convert back to radical notation if needed.

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. This concept allows for the simplification of expressions involving roots and powers, making it easier to manipulate algebraic expressions. Understanding how to convert between radical and exponent notation is crucial for solving problems involving roots.

Radical Notation

Radical notation is a mathematical notation used to denote roots of numbers or expressions. The symbol √ (the radical sign) indicates the square root, while other roots are represented with an index, such as ³√ for cube roots. Converting expressions from rational exponents to radical notation is often required in algebra to present answers in a more recognizable form, especially when dealing with roots.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, which often includes combining like terms, factoring, and applying the properties of exponents. In the context of rational exponents, this means using the rules of exponents to combine terms effectively. Mastery of simplification techniques is essential for solving algebraic problems accurately and efficiently.