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.___³√8a⁶

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1. Rewrite the cube root expression using rational exponents: \( \sqrt[3]{8a^6} = (8a^6)^{1/3} \).
2. Apply the property of exponents \((ab)^n = a^n b^n\) to separate the expression: \((8)^{1/3} \cdot (a^6)^{1/3}\).
3. Simplify \((8)^{1/3}\) by recognizing that 8 is a perfect cube: \((8)^{1/3} = 2\).
4. Simplify \((a^6)^{1/3}\) using the power of a power property \((a^m)^n = a^{m \cdot n}\): \(a^{6 \cdot (1/3)} = a^2\).
5. Combine the simplified parts: \(2a^2\).

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 concept allows for the simplification of expressions involving roots and powers, making calculations more manageable.

Radical Notation

Radical notation is a mathematical notation used to denote roots. The expression √x represents the square root of x, while n√x denotes the n-th root of x. Understanding how to convert between radical notation and rational exponents is essential for simplifying expressions and solving equations involving roots.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, often by combining like terms or applying exponent rules. In the context of rational exponents and radicals, this may include rewriting expressions to eliminate complex fractions or roots, making them easier to work with in further calculations.