Problem 63 - Intermediate Algebra

Question

Simplify by combining like radicals. $$ 14 \sqrt[4]{32}-15 \sqrt[4]{2} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 13 \sqrt[4]{2} \).

Step 1: Simplify the Radicals

First, simplify each radical expression. Start with the term \( \sqrt[4]{32} \). Notice that 32 can be written as \( 2^5 \). Since we are looking for the fourth root, we simplify it to \( 2 \cdot \sqrt[4]{2} \). Therefore, \( 14 \sqrt[4]{32} \) simplifies to \( 14 \cdot 2 \sqrt[4]{2} = 28 \sqrt[4]{2} \).

Step 2: Recollect the Terms

Now that we have simplified \( 14 \sqrt[4]{32} \) to \( 28 \sqrt[4]{2} \), rewrite the expression using this simplified form: \( 28 \sqrt[4]{2} - 15 \sqrt[4]{2} \).

Step 3: Combine Like Terms

Both terms involve \( \sqrt[4]{2} \). So, simply subtract the coefficients: \( 28 - 15 = 13 \). Therefore, the expression becomes \( 13 \sqrt[4]{2} \).

Key Concepts

Combining Like Radicals

Combining like radicals is similar to combining like terms in algebra. Just as you can add or subtract terms with the same variable, you can do the same with radicals that have the same root and the same radicand. This approach makes the simplification process much easier and organized.

To successfully combine like radicals:

• Ensure that the radicals have the same index. In our case, both terms are fourth roots.
• Check that the radicands (the numbers under the root) are identical. Here, both terms have a radicand of 2.

Once these conditions are satisfied, focus on the coefficients, as these are the numbers you'll add or subtract. For example, if you have expressions like \( 28 \sqrt[4]{2} \) and \( 15 \sqrt[4]{2} \), treat the radical part as a common factor. You can subtract the coefficients, resulting in \( 13 \sqrt[4]{2} \).

Remember, you can't combine radicals that don't meet these matching criteria. Each type of radical expression you'll encounter needs this kind of careful inspection and manipulation to simplify successfully.

Fourth Roots

Understanding fourth roots is crucial when simplifying radical expressions, especially when dealing with higher-order roots than the more common square roots. The fourth root of a number \( x \) is a value that, when multiplied by itself four times, yields \( x \). Mathematically, this can be expressed as \( \sqrt[4]{x} \) or \( x^{1/4} \).

To simplify a fourth root, break down the radicand into its prime factors. For instance, when dealing with \( \sqrt[4]{32} \):

• Recognize that 32 is \( 2^5 \).
• Identify how many complete sets of 4 are in the factorization (here, there's one set of \( 2^4 \)).
• For each complete set, take one of those numbers outside the root. \( 2 \cdot \sqrt[4]{2} \).

This process helps in translating the radical into a product involving whole numbers, which is often more straightforward to work with in combination expressions like the one in our original problem.

Like Terms

In algebra and radicals, 'like terms' are terms that contain the same variable raised to the same power or the same part under a radical. Recognizing and combining like terms simplifies expression and makes it easier to work with and understand.

Take the example from our exercise: \( 28 \sqrt[4]{2} - 15 \sqrt[4]{2} \). Here, the radical part, \( \sqrt[4]{2} \), is identical. This allows us to directly operate with their coefficients, making simplification intuitive.

• The process involves aligning and only combining coefficients of these roots, similar to algebra.
• Ensure that you're combining based on both the identical root and matching radicands.

When radicals or terms in an expression don't match this way, they cannot be combined. Being able to identify and work with like terms is fundamental for solving equations and simplifying expressions efficiently in math.