Problem 56 - Algebra for College Students

Question

Express each of the following in simplest radical form. All variables represent positive real numbers. \(\sqrt[3]{\frac{5}{2 x}}\)

Step-by-Step Solution

Answer

\( \frac{\sqrt[3]{5}}{\sqrt[3]{2x}} \) is the simplest radical form.

Step 1: Identify the Expression

Our expression is \( \sqrt[3]{\frac{5}{2x}} \), which is a cube root expression. We want to simplify this expression to its simplest radical form.

Step 2: Apply Cube Root to Fraction

To simplify \( \sqrt[3]{\frac{5}{2x}} \), recognize that the cube root can be applied to both the numerator and the denominator separately. This gives us \( \frac{\sqrt[3]{5}}{\sqrt[3]{2x}} \).

Step 3: Simplify the Denominator Cube Root

Further simplify the denominator \( \sqrt[3]{2x} \). Note that \( \sqrt[3]{2x} = \sqrt[3]{2} \times \sqrt[3]{x} \).

Step 4: Combine the Simplified Parts

Putting it all together, the expression becomes \( \frac{\sqrt[3]{5}}{\sqrt[3]{2} \times \sqrt[3]{x}} \).

Step 5: Write the Final Simplified Form

The simplest radical form is \( \frac{\sqrt[3]{5}}{\sqrt[3]{2} \cdot \sqrt[3]{x}} \) as we have separated the components into simpler radicals.

Key Concepts

Cube Roots

Imagine a cube with all sides of equal length. The cube root of a number seeks to find the length of those sides if the entire volume is given. For example, the cube root of 8 is 2, because if each side of a cube is 2, the volume is indeed 8.In algebra, finding the cube root means determining a number which, when multiplied by itself twice, returns the original number. The cube root of a number or expression is denoted as \( \sqrt[3]{\ldots} \).

• If you have \( \sqrt[3]{27} \), the answer is 3, since 3 x 3 x 3 = 27.
• In expressions like \( \sqrt[3]{\frac{5}{2x}} \), we apply the cube root separately to the numerator and the denominator.

Cube roots are essential in breaking down complex expressions into more manageable pieces, enabling us to simplify and solve various algebraic problems.

Simplest Radical Form

Transforming expressions into their simplest radical form involves simplifying the radicals as much as possible. This means separating an expression into its primary roots without any radicals in the denominator.

• For fractions, this includes simplifying each part of the fraction independently, such as \( \sqrt[3]{5} \) and \( \sqrt[3]{2x} \).
• Each component of the expression is handled separately to achieve a cleaner, simpler look.

The goal is to make the expression as straightforward as it can possibly be. No further breakdown is possible after reaching its simplest radical form. In the example, \( \frac{\sqrt[3]{5}}{\sqrt[3]{2} \cdot \sqrt[3]{x}} \) is simpler because the radicals are easy to identify and work with. Keeping expressions clean and uncomplicated helps in solving equations and performing other mathematical operations efficiently.

Algebraic Expressions

Algebraic expressions are the building blocks of algebra. They are combinations of numbers, variables, and operations that stand in for specific values or quantities. Understanding how to manipulate these expressions is critical in simplifying and solving problems.

• In expressions like \( \sqrt[3]{\frac{5}{2x}} \), the numbers (5 and 2) and the variable (x) are tied together using operations in the fraction.
• These expressions can represent real-world problems, such as dynamic measurements and changes in quantities.

Handling algebraic expressions often involves employing techniques like factoring, distributing, and simplification. By transforming an expression into its simplest form, like separating cube roots, we make it easier to handle. The clarity gained from a simplified expression aids in further mathematical exploration and understanding.