Problem 103 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. All variables represent positive real numbers. $$ \sqrt[3]{24 x}+\sqrt[3]{3 x} $$

Step-by-Step Solution

Answer

The simplified expression is \(3\sqrt[3]{3x}\).

Step 1: Factor Inside the Cube Roots

First, notice that both terms inside the cube roots can be factored to reveal common factors. We will factor each term separately:\[24x = (8 \times 3)x = 8x \times 3\quad \text{and}\quad 3x = 3x \times 1.\]

Step 2: Break Down the Cube Roots

Next, express each term inside their respective cube roots:\[\sqrt[3]{24x} = \sqrt[3]{8x \cdot 3} = \sqrt[3]{8x} \cdot \sqrt[3]{3}\]and\[\sqrt[3]{3x} = \sqrt[3]{3x \cdot 1} = \sqrt[3]{3x}\]

Step 3: Evaluate Simple Cube Roots

Since \(8x\) is a perfect cube in terms of \(8\), simplify its cube root:\[\sqrt[3]{8x} = \sqrt[3]{8} \cdot \sqrt[3]{x} = 2 \cdot \sqrt[3]{x}\]

Step 4: Combine Like Terms

Now, we have terms that have \(\sqrt[3]{x}\) in common:\[2\sqrt[3]{x}\sqrt[3]{3} + \sqrt[3]{3x}= 2\sqrt[3]{3x} + \sqrt[3]{3x}\]Combine them:\[3\sqrt[3]{3x}\]

Step 5: Simplified Expression Result

Thus, the simplified form of the expression is \(3\sqrt[3]{3x}\).

Key Concepts

Understanding Cube Roots

Cube roots are an important mathematical concept that allow us to find a special value that, when multiplied by itself three times, returns the original number. For any number or variable "a," the cube root is represented as \(\sqrt[3]{a}\). This is different from the square root, which involves multiplication only twice.\
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When simplifying cube roots, identifying perfect cubes helps. Perfect cubes are numbers like 1, 8, 27, etc., because they can be expressed as \(n^3\), where "n" is an integer. In the given exercise, 8 is a perfect cube, which simplifies the cube root of 24x, as we see it can be factored to include 8.\
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Breaking numbers down to find the cube root of each factor makes it easier. Like in the expression \(\sqrt[3]{24x} = \sqrt[3]{8x} \cdot \sqrt[3]{3}\).\
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This shows the process of looking for elements within an expression that can be simplified using well-known perfect cubes.

Factoring in Algebraic Simplification

Factoring is a key skill in algebra that involves breaking down numbers or expressions into products of simpler 'factors.' This makes solving equations and simplifying expressions much more manageable.\
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For the provided exercise, to tackle the cube roots, we first factor each expression under the root. For example, 24x is factored into \(8 \times 3 \times x\), highlighting that 8 is a perfect cube and simplifies easily.\
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Here’s why factoring is useful:

• It reveals hidden perfect cubes that simplify roots.
• It helps identify common factors.
• It breaks down complex expressions into simpler parts.

\Each step in the calculation undeniably shows the power of this technique, allowing the expression to be rewritten as simpler terms.

Combining Like Terms

Combining like terms is an algebraic technique used to simplify expressions by merging terms that have the same variable parts. This often reduces the expression to a much simpler form, making calculations easier.\
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In the example provided, after simplifying, both \(2\sqrt[3]{3x}\) and \(\sqrt[3]{3x}\) have \(\sqrt[3]{3x}\) in common. This indicates they are 'like terms.'\
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To combine them, notice how similar parts can be added like simple integers: \(2 + 1 = 3\). Therefore, the expression becomes \(3\sqrt[3]{3x}\).\
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Let's break it down:

• Identify terms with the same variable factors.
• Add their coefficients while keeping the variable part unchanged.
• Simplify the expression to its simplest form.

\Using this technique helps in managing and simplifying algebraic expressions efficiently.