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The fourth root of a number is a special type of radical expression. It represents the number that, when multiplied by itself four times, equals the original number. For example, in the expression \( \sqrt[4]{32} \), you are looking for a number that, when raised to the fourth power, would give you 32. Understanding how to calculate fourth roots is essential for tackling radical expressions.

To break down \( \sqrt[4]{32} \), we start by factoring the number 32. This gives us \( 32 = 2^5 \). Taking the fourth root of \( 2^5 \) involves separating the exponents: \( 2^5 \) can be rewritten as \( (2^4) \times 2 \). This means \( \sqrt[4]{32} = \sqrt[4]{(2^4 \times 2)} = 2 \sqrt[4]{2} \).

A similar approach is applied to \( \sqrt[4]{128} \), where we factor 128 as \( 2^7 \), and calculate its fourth root accordingly. Understanding this technique of breaking down numbers is key to simplifying radical expressions.

Simplifying radical expressions involves reducing them to their simplest form, which is one of the main objectives in algebra. To simplify a radical like \( \sqrt[4]{32} \), it involves expressing the number under the root as a power of primes. This allows us to extract factors that are perfect powers.

In general:

• Look for perfect powers under the root sign
• Factor numbers systematically using prime factorization
• Directly utilize perfect powers by taking clean roots

Step by step, simplification makes complex calculations more manageable and easy to understand. For \( \sqrt[4]{128} \), you factor 128, simplify it to \( 2 \sqrt{2} \), and similarly combine such expressions to their simplest form whenever possible. Recognizing patterns can help simplify complex expressions effectively, as seen in the original problem.

The addition of radicals involves combining like terms, which can sometimes be simplified further. The radicals must have the same radicand and index for direct addition or subtraction.

In the exercise \( \sqrt[4]{32} + \sqrt[4]{128} \), once each radical is simplified, you can add the coefficients if the radicals are similar. However, in some cases, such as this one, after simplification, the radical parts may not match up perfectly. The remaining expression is already simplified if no further like terms are present.

• Check if the radicals have the same index and radicand before combining
• Add only the coefficients of like radicals
• Always look for further simplification possibilities

Understanding when and how to combine radicals is a vital part of working with radical expressions, leading to a neater and more comprehensible result.