Problem 69 - Elementary and Intermediate Algebra

Question

Simplify by combining like radicals. All variables represent positive real numbers. $$ 2 \sqrt[3]{16}-\sqrt[3]{54}-3 \sqrt[3]{128} $$

Step-by-Step Solution

Answer

The simplified expression is \(-11 \sqrt[3]{2}\).

Step 1: Simplify the Radicals

First, we break down each of the radicals to their simplest form.1. For \(\sqrt[3]{16}\): \[\sqrt[3]{16} = \sqrt[3]{2^4} = \sqrt[3]{2^3 \times 2} = 2 \cdot \sqrt[3]{2}\] 2. For \(\sqrt[3]{54}\): \[\sqrt[3]{54} = \sqrt[3]{2 \times 3^3} = 3 \cdot \sqrt[3]{2}\] 3. For \(\sqrt[3]{128}\): \[\sqrt[3]{128} = \sqrt[3]{2^7} = \sqrt[3]{(2^3)^2 \times 2} = 4 \cdot \sqrt[3]{2}\]

Step 2: Substitute Simplified Radicals

Now substitute the simplified values back into the original expression:\[2 \cdot \sqrt[3]{16} - \sqrt[3]{54} - 3 \cdot \sqrt[3]{128} \rightarrow 2(2 \cdot \sqrt[3]{2}) - (3 \cdot \sqrt[3]{2}) - 3(4 \cdot \sqrt[3]{2})\]

Step 3: Distribute and Combine Like Terms

Distribute the coefficients and then combine like terms.The expression becomes:- \(2 \cdot 2 \cdot \sqrt[3]{2} = 4 \sqrt[3]{2}\)- \(-3 \cdot \sqrt[3]{2} = -3 \sqrt[3]{2}\)- \(-3 \cdot 4 \cdot \sqrt[3]{2} = -12 \sqrt[3]{2}\)Combine these: \[4 \sqrt[3]{2} - 3 \sqrt[3]{2} - 12 \sqrt[3]{2} = -11 \sqrt[3]{2}\]

Step 4: Final Answer

We have now simplified the expression by combining like radicals. The final answer is: \[-11 \sqrt[3]{2}\].

Key Concepts

Combining Like Terms

The expression given requires the process of combining like terms to simplify it. When you are working with algebraic expressions, "like terms" are terms that have the same variable raised to the same power. For example, in our exercise, all parts of the expression ultimately relate to terms involving \(\sqrt[3]{2}\). This similarity allows us to combine them into a single term.

• First, identify the like terms. These are the terms that have the same root part. In the problem, after simplifying the radicals, each term shares \(\sqrt[3]{2}\).
• Next, add or subtract these like terms by combining their coefficients. In the problem, the coefficients 4, -3, and -12 can be combined, since the root is the same: \(4 \sqrt[3]{2} - 3 \sqrt[3]{2} - 12 \sqrt[3]{2} = -11 \sqrt[3]{2}\).

This process simplifies the expression to its most straightforward form. Remember, combining like terms where the radicals are similar is crucial for simplifying radical expressions. This ensures that the term is expressed as neatly and simply as possible.

Cube Roots

Cube roots are an essential concept in simplifying radical expressions, especially when dealing with expressions where the root is common among terms. In the exercise, we work with cube roots (\(\sqrt[3]{x}\)). Understanding these helps in breaking down complex numbers into more manageable forms.

• Cube root, denoted by \(\sqrt[3]{x}\), refers to a number which when multiplied by itself three times results in \(x\). For instance, \(\sqrt[3]{8} = 2\) because \(2 \times 2 \times 2 = 8\).
• In our exercise, recognizing cube roots helps in expressing powers of numbers and simplifying expressions. For example, \(\sqrt[3]{16} = 2\cdot\sqrt[3]{2}\), because \(16 = 2^4\) and \(\sqrt[3]{2^3 \times 2} = 2 \cdot \sqrt[3]{2}\).

Working with cube roots can unlock more straightforward versions of otherwise complex expressions. It's crucial to fully factor the number under the radical to employ cube roots effectively.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. This concept is pivotal in simplifying expressions involving root calculations. In our exercise, the manipulating of cube roots to achieve like radicals was a critical procedure.

• To simplify a radical expression, write the number inside the radical as a product of its prime factors. For example, turning \(54\) into \(2 \times 3^3\).
• Extract the perfect powers from the radical. If it's a perfect cube, as \(3^3\) in \(\sqrt[3]{54}\), you extract \(3\).
• After breaking down the expression, like radicals can be simplified using combining like terms.

By working through these steps, radical expressions become less daunting and more approachable. It revolves around recognizing the components within a radical and manipulating them to simplify the expression efficiently.