Problem 67 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ 2 \sqrt[3]{6}-7 \sqrt[3]{6} $$

Step-by-Step Solution

Answer

The expression simplifies to \(-5 \sqrt[3]{6}\).

Step 1: Identify Like Terms

Observe the expression: \(2 \sqrt[3]{6} - 7 \sqrt[3]{6}\). Notice that both terms involve the cube root of 6, indicated by \(\sqrt[3]{6}\). These are like terms, which means they can be combined by adding or subtracting their coefficients.

Step 2: Combine Coefficients

Since \(\sqrt[3]{6}\) is a common factor, factor it out from the expression: \((2 - 7) \sqrt[3]{6}\). Simplify the coefficients: \(2 - 7 = -5\).

Step 3: Simplify the Expression

Substitute the simplified coefficient back into the expression: \(-5 \sqrt[3]{6}\). This is the simplified form of the original expression.

Key Concepts

Like Terms

When working with algebraic expressions, it's essential to identify like terms. Like terms are components of an expression that have the same variable part raised to the same power. In some cases, these can include similar radical parts. The key is that they have identical variables and exponents or radicals.
In our example, the terms involve cube roots. Specifically, the terms are: \(2 \sqrt[3]{6}\) and \(-7 \sqrt[3]{6}\). Both include the cube root, \(\sqrt[3]{6}\), and are therefore like terms.

• Identifying like terms is crucial because it allows us to combine them to simplify expressions easily.
• Look for common variables, exponents, or radical components when identifying like terms.

This simplification is possible because the terms share a common factor, \(\sqrt[3]{6}\), allowing us to directly add or subtract their coefficients to streamline the expression.

Coefficients

In any algebraic term, the coefficient is the numerical factor that is multiplied by the variable part of the term. This number tells us how many times we are taking that variable or radical factor. In the expression \(2 \sqrt[3]{6} - 7 \sqrt[3]{6}\), the coefficients are 2 and -7 respectively.
The coefficient acts as an indicator of the term's size and direction:

• Positive or negative: determines whether the term adds to or subtracts from the sum.
• Magnitude: the absolute value of the coefficient tells us how many times the variable factor is taken.

In the simplification process, we focused on these coefficients. We subtracted 7 from 2 because the like terms \(\sqrt[3]{6}\) allowed us to combine the coefficients: \[2 - 7 = -5\]This results in a new coefficient of -5, simplifying the expression to \(-5 \sqrt[3]{6}\). Understanding coefficients is key to manipulating and simplifying algebraic expressions.

Cube Roots

Cube roots are a specific type of radical that tells us what number, when multiplied by itself three times, gives the original number. The cube root of a number \(x\) is written as \(\sqrt[3]{x}\). For our particular example with \(\sqrt[3]{6}\), it signifies what number cubed equals 6.
Cube roots differ from square roots in that they deal with cubing, or raising to the power of three, rather than squaring.

• Cube roots can be positive or negative because both positive and negative numbers raised to an odd power yield the original number.
• If you have a cube root in your expression, it means you can combine terms involving the same cube root easily.

In our expression, we treated \(\sqrt[3]{6}\) as the common factor for like terms. This allowed us to streamline the expression by merely focusing on the coefficients, leading to an elegant solution: \(-5 \sqrt[3]{6}\). Understanding cube roots thus aids in recognizing patterns and simplifying expressions effectively.