Problem 76 - Algebra and Trigonometry

Question

Add or subtract terms whenever possible. $$ 6 \sqrt[5]{3}+2 \sqrt[5]{3} $$

Step-by-Step Solution

Answer

The simplified form of \( 6 \sqrt[5]{3} + 2 \sqrt[5]{3} \) is \( 8 \sqrt[5]{3} \).

Step 1: Identify the Like Terms

First, identify the like terms in the expression. In this case, \( 6 \sqrt[5]{3} \) and \( 2 \sqrt[5]{3} \) are like terms because they have the same variable part which is \( \sqrt[5]{3} \).

Step 2: Add/Subtract the Like Terms

Then add these like terms. When adding like terms, add just the coefficients. The coefficient of \( \sqrt[5]{3} \) in \( 6 \sqrt[5]{3} \) is 6 and in \( 2 \sqrt[5]{3} \) is 2. Adding 6 and 2 gives 8.

Step 3: Write the Result

Finally, write down the result by attaching the common variable part to the result of the addition. So, \( 6 \sqrt[5]{3} + 2 \sqrt[5]{3} = 8 \sqrt[5]{3} \).

Key Concepts

Coefficient

The concept of coefficients in mathematics may initially seem daunting, but it's actually quite straightforward. A coefficient is essentially the numeric part of any term, especially when working with expressions involving variables or radicals. For example, in the radical expression \( 6 \sqrt[5]{3} \), the coefficient is the number 6. It tells us how many "units" of the radical \( \sqrt[5]{3} \) we have.

In contexts involving radicals, the coefficient plays a crucial role in addition and subtraction. Suppose you wish to simplify an expression involving

• \( 6 \sqrt[5]{3} \)
• \( 2 \sqrt[5]{3} \)

You would focus on the coefficients—6 and 2. These numbers are treated much like regular numbers in addition. By adding them together, as you would with integers, you find the sum of the coefficients. In this case, adding 6 and 2 gives you 8, simplifying the overall expression to \( 8 \sqrt[5]{3} \).

Understanding coefficients is essential in simplifying radical expressions, as it allows you to combine terms efficiently without dealing with the complexity of the radicals themselves.

Like Terms

"Like terms" is a fundamental concept in algebra that helps in simplifying expressions. In simpler terms, like terms are terms that have the same variables raised to the same power. For radicals, like terms must have the same radical part.

Taking the example from our original problem, consider:

• \( 6 \sqrt[5]{3} \)
• \( 2 \sqrt[5]{3} \)

These are like terms because both terms include the exact same radical, \( \sqrt[5]{3} \). Although the numbers in front (coefficients) differ, the radical component they are multiplying is identical.

Recognizing like terms is an essential skill for simplifying radicals. Only like terms can be added or subtracted directly. Once identified, you simply perform arithmetic on the coefficients while keeping the radical part unchanged.

Addition of Radicals

Adding radicals follows rules similar to those for adding variables or numerical terms, but it also has unique considerations. For the addition to occur, radicals must be like terms, meaning their radicands (the numbers inside the radical) and their indices (the root power) must be identical.

Consider the expression:

• \( 6 \sqrt[5]{3} + 2 \sqrt[5]{3} \)

Here, both radicals are \( \sqrt[5]{3} \), allowing us to sum the coefficients: 6 and 2. Thus, the operation becomes a simple addition of the coefficients, while the radical part \( \sqrt[5]{3} \) remains constant, resulting in

\[ 6 \sqrt[5]{3} + 2 \sqrt[5]{3} = 8 \sqrt[5]{3} \]

However, if the radicals differ, such as \( \sqrt[3]{3} \) and \( \sqrt[5]{3} \), they cannot be added together directly. They are not like terms due to differing indices or radicands, and thus must remain separate in any simplified expression.

This rule ensures that the simple process of combining coefficients is not disrupted by differing radical parts.