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√x⁵√x

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1. Step 1: Express each radical using rational exponents. The expression \( \sqrt[4]{x} \) can be written as \( x^{1/4} \) and \( \sqrt[5]{x} \) can be written as \( x^{1/5} \).
2. Step 2: Multiply the expressions with rational exponents. This means multiplying \( x^{1/4} \) and \( x^{1/5} \).
3. Step 3: Use the property of exponents that states \( a^m \cdot a^n = a^{m+n} \). Therefore, add the exponents: \( x^{1/4 + 1/5} \).
4. Step 4: Find a common denominator to add the fractions \( \frac{1}{4} \) and \( \frac{1}{5} \). The common denominator is 20.
5. Step 5: Rewrite the sum of the exponents as a single fraction: \( x^{\frac{5}{20} + \frac{4}{20}} = x^{\frac{9}{20}} \).

Key Concepts

Rational Exponents

Rational exponents are a way to express roots using fractional powers. For example, the expression x^(1/n) represents the n-th root of x. This notation allows for easier manipulation of expressions involving roots, as it can be combined with other algebraic operations like multiplication and division.

Radical Notation

Radical notation is a mathematical way to represent roots, using the radical symbol (√). For instance, the square root of x is written as √x, while the n-th root is expressed as n√x. Understanding how to convert between radical and rational exponent forms is essential for simplifying expressions effectively.

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). Mastery of these properties is crucial for simplifying expressions with rational exponents.