Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. ∜∛2

Verified step by step guidance

1. Step 1: Recognize that the expression involves two different roots: a fourth root (∜) and a cube root (∛).
2. Step 2: Rewrite the expression using rational exponents. The fourth root of a number can be expressed as raising the number to the power of 1/4, and the cube root as raising to the power of 1/3.
3. Step 3: Express the given expression ∜∛2 as (2^(1/3))^(1/4).
4. Step 4: Apply the power of a power property of exponents, which states that (a^m)^n = a^(m*n).
5. Step 5: Multiply the exponents: (1/3) * (1/4) to simplify the expression to 2 raised to the power of the resulting exponent.

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and higher-order roots. The notation ∜ and ∛ indicates the fourth root and cube root, respectively. Understanding how to manipulate these expressions is essential for performing operations like addition, subtraction, multiplication, and division involving radicals.

Properties of Exponents

The properties of exponents are rules that govern how to simplify expressions involving powers. For instance, the rule a^(m/n) represents the n-th root of a raised to the m-th power. This concept is crucial when dealing with radical expressions, as it allows for the conversion between radical and exponential forms, facilitating easier calculations.

Simplifying Radicals

Simplifying radicals involves reducing radical expressions to their simplest form. This includes factoring out perfect squares, cubes, or higher powers from under the radical sign. Mastery of this concept is important for accurately performing operations on radicals, ensuring that the final answer is expressed in the most concise and manageable way.