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

Verified step by step guidance

1. Recognize that the expression involves fourth roots (also called fourth radicals). The symbol ∜x means the fourth root of x, which can be written as \(x^{\frac{1}{4}}\).
2. Rewrite each term using fractional exponents: \(\sqrt[4]{32} = 32^{\frac{1}{4}}\) and \(3\sqrt[4]{2} = 3 \times 2^{\frac{1}{4}}\).
3. Factor the expression to see if the terms have a common radical factor. Notice that both terms involve \(2^{\frac{1}{4}}\), since 32 can be expressed as a power of 2: \$32 = 2^5$.
4. Rewrite \(32^{\frac{1}{4}}\) as \((2^5)^{\frac{1}{4}} = 2^{\frac{5}{4}}\). Now the expression is \(2^{\frac{5}{4}} + 3 \times 2^{\frac{1}{4}}\).
5. Factor out the smaller power of \(2^{\frac{1}{4}}\) from both terms: \(2^{\frac{1}{4}} (2^{1} + 3)\). Simplify inside the parentheses to complete the factoring step.

Key Concepts

Radical Expressions and Roots

Radical expressions involve roots such as square roots, cube roots, and fourth roots (∜). Understanding how to interpret and simplify these roots is essential, especially recognizing that ∜32 means the fourth root of 32, which can be rewritten using exponents or prime factorization.

Simplifying Radicals

Simplifying radicals involves expressing the radicand (the number inside the root) as a product of perfect powers and other factors. This helps to extract factors outside the radical, making addition or other operations easier. For example, breaking down 32 into powers of 2 helps simplify ∜32.

Operations with Radicals

Adding radicals requires that the radicals have the same index and radicand. If they differ, you must simplify them first or express them with a common radical form. In this problem, understanding how to combine ∜32 and 3∜2 involves simplifying and possibly rewriting the radicals to determine if they can be combined.