Perform the indicated operations. Assume all variables represent positive real numbers. ∜32 + 3∜2

Verified step by step guidance

1. Step 1: Identify the terms involved in the expression. The expression is ∜32 + 3∜2, which involves two terms: ∜32 and 3∜2.
2. Step 2: Simplify the first term, ∜32. The fourth root of 32 can be expressed as 32^{1/4}.
3. Step 3: Simplify the second term, 3∜2. The term 3∜2 can be expressed as 3 times the fourth root of 2, or 3 * 2^{1/4}.
4. Step 4: Consider simplifying further by expressing both terms with the same base if possible. Check if 32 can be expressed in terms of 2, since 32 = 2^5.
5. Step 5: Combine the terms if possible, by expressing them with a common base or simplifying further if they have like terms.

Key Concepts

Radicals

Radicals are expressions that involve roots, such as square roots, cube roots, and higher-order roots. In this question, the fourth root (∜) is used, which means finding a number that, when raised to the fourth power, equals the given number. Understanding how to simplify and manipulate radical expressions is essential for performing operations involving them.

Exponents

Exponents represent repeated multiplication of a number by itself. In the context of radicals, exponents can be used to express roots in fractional form, where the denominator indicates the root's order. For example, the fourth root of a number can be expressed as raising that number to the power of 1/4. This concept is crucial for simplifying expressions involving radicals.

Combining Like Terms

Combining like terms is a fundamental algebraic skill that involves simplifying expressions by adding or subtracting terms that have the same variable parts. In the expression ∜32 + 3∜2, it is important to recognize that these terms are not like terms due to their different radicands, which means they cannot be combined directly. Understanding this concept helps in correctly simplifying and evaluating expressions.