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√√x

Verified step by step guidance

1. Start by expressing the innermost radical, \( \sqrt{x} \), using rational exponents. Recall that \( \sqrt{x} = x^{1/2} \).
2. Next, express the outer radical, \( \sqrt[4]{\sqrt{x}} \), using rational exponents. This becomes \( (x^{1/2})^{1/4} \).
3. Apply the power of a power property of exponents, which states that \((a^m)^n = a^{m \cdot n}\). Here, multiply the exponents: \( \frac{1}{2} \times \frac{1}{4} \).
4. Simplify the multiplication of the exponents to get a single rational exponent.
5. Finally, convert the expression back to radical notation, if necessary, by using the simplified rational exponent.

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 exponent rules. Understanding how to convert between radical and rational exponent forms is essential for simplifying expressions.

Radical Notation

Radical notation is a mathematical notation used to denote roots, such as square roots or cube roots. The radical symbol (√) indicates the root of a number, where the index of the root is specified if it is not a square root. For instance, √x represents the square root of x, while ⁴√x denotes the fourth root. Converting between radical and rational exponent forms is crucial for expressing simplified results.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, making them easier to work with. This process often includes combining like terms, applying exponent rules, and converting between different forms of notation. In the context of rational exponents and radicals, simplification may require rewriting expressions to eliminate complex roots or fractional exponents, ensuring clarity and ease of interpretation.