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.__⁸√x²

Verified step by step guidance

1. Rewrite the expression using rational exponents: \( \sqrt[8]{x^2} = x^{\frac{2}{8}} \).
2. Simplify the exponent by reducing the fraction: \( x^{\frac{2}{8}} = x^{\frac{1}{4}} \).
3. Convert the expression back to radical notation: \( x^{\frac{1}{4}} = \sqrt[4]{x} \).

Key Concepts

Rational Exponents

Rational exponents are a way to express roots using fractional powers. For example, the expression a^(m/n) represents the n-th root of a raised to the m-th power. This notation allows for easier manipulation of expressions involving roots and powers, making it essential for simplifying expressions like the one given.

Radical Notation

Radical notation is a mathematical way to express roots, such as square roots or cube roots, using the radical symbol (√). For instance, the expression √a is equivalent to a^(1/2). Understanding how to convert between radical notation and rational exponents is crucial for simplifying expressions and presenting answers in the required format.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, which often includes combining like terms, factoring, and applying exponent rules. In the context of rational exponents, this means rewriting expressions to eliminate complex fractions or roots, making it easier to interpret and solve mathematical problems.