Question

Combine radicals, if possible. \(\sqrt[3]{4 x}+6 \sqrt[3]{4 x}-2 \sqrt[3]{4 x}\)

Short answer

Question: Simplify the expression \(\sqrt[3]{4x} + 6\sqrt[3]{4x} - 2\sqrt[3]{4x}\). Answer: \(5\sqrt[3]{4x}\)

Step-by-step solution

Identify the Common Term

The common term among the given terms is \(\sqrt[3]{4x}\).

Apply Arithmetic Operations to Combine the Terms

Since the common term is \(\sqrt[3]{4x}\), we can add/subtract their coefficients. The given expression is: \(\sqrt[3]{4 x}+6 \sqrt[3]{4 x}-2 \sqrt[3]{4 x}\) The coefficients of \(\sqrt[3]{4x}\) are 1, 6, and -2. We can add the coefficients: \(1 + 6 - 2 = 5\)

Write the Final Expression

After adding the coefficients, we can rewrite the expression as: \(5\sqrt[3]{4x}\) The simplified expression combining the given terms is \(5\sqrt[3]{4x}\).

Key concepts

Simplifying Expressions

Simplifying expressions involves making an equation or mathematical phrase easier to interpret. When you simplify, you reduce the number of terms or factors into its simplest form. This process often includes combining like terms, which are terms that contain the same variables raised to the same power. Here, we're dealing with radicals, specifically cube roots, and our goal is to combine them by looking closely at any shared components they might have.

• Identify like terms, which in this context are radicals with the same base and index.
• Use basic arithmetic operations to simplify.

This process helps to express the function more compactly, making it easier to understand and use in further calculations. By identifying and combining common terms, the expression is condensed, making complex calculations more manageable and mistakes less likely.

Cube Roots

Cube roots are special types of radicals where the root degree is three. They are used to determine what number, when multiplied by itself three times, yields the original number. For example, \(\sqrt[3]{8} = 2 \) because \( 2 \times 2 \times 2 = 8 \). Cube roots can be more challenging to simplify than square roots, but the approach remains the same: identifying common factors.

• Recognize the need for the cube root sign \(\sqrt[3]{ }\).
• Understand that cube roots can often be combined or simplified when they have the same radicand.

In solving expressions with cube roots, it's crucial to capture any shared values, which assists in simplification. This was evident in the original problem where \(\sqrt[3]{4x}\) was a shared term across all items. By accounting for this commonality, simplification becomes much more straightforward.

Arithmetic Operations in Algebra

Arithmetic operations in algebra include addition, subtraction, multiplication, and division. These operations are crucial in simplifying expressions and solving equations. When dealing with expressions like those that include cube roots, you apply these arithmetic operations to the coefficients of the like terms. Here's a closer look at how it was done in the original problem:

• Identify like terms in the expression.
• Add or subtract these terms by focusing on their coefficients, as done with \(1 + 6 - 2 = 5\).

This procedure, where you add or subtract the coefficients of similar terms, allows the expression to be written in a more consolidated form. These basic arithmetic skills provide the fundamental tools necessary for more complex algebraic manipulations and are pivotal in ensuring accuracy and clarity in mathematical computations.