Problem 6 - A Graphical Approach to College Algebra

Question

Match the rational exponent expression in Exercises \(1-8\) with the equivalent radical expression in \(A-H\). Assume that \(x \neq 0\). A. \(\frac{3}{\sqrt[3]{x}}\) B. \(-3 \sqrt[3]{x}\) C. \(\frac{1}{\sqrt[3]{3 x}}\) D. \(\frac{-3}{\sqrt[3]{x}}\) E. \(3 \sqrt[3]{x}\) F. \(\sqrt[3]{-3 x}\) G. \(\sqrt[3]{3 x}\) H. \(\frac{1}{\sqrt[3]{-3 x}}\) $$3 x^{-1 / 3}$$

Step-by-Step Solution

Answer

The equivalent radical expression is A.

Step 1: Understand the Expression

The given expression is \(3x^{-1/3}\). This means you need to understand how to handle rational exponents.

Step 2: Convert the Rational Exponent to a Radical

The expression \(x^{-1/3}\) can be rewritten as \(\frac{1}{x^{1/3}}\). This means we are dealing with the cube root of \(x\), or \(\sqrt[3]{x}\). Therefore, \(x^{-1/3} = \frac{1}{\sqrt[3]{x}}\).

Step 3: Incorporate the Coefficient

The entire expression becomes \(3 \cdot \frac{1}{\sqrt[3]{x}}\), which simplifies to \(\frac{3}{\sqrt[3]{x}}\).

Step 4: Match with Options

Look through the provided options (A-H) for an equivalent expression. Option A, which is \(\frac{3}{\sqrt[3]{x}}\), matches the expression we derived.

Key Concepts

Radical Expressions

Radical expressions involve roots of numbers or variables, like square roots or cube roots. These expressions can often look intimidating, but they're essentially about finding a number that, when multiplied by itself a certain number of times, will give you the original number.
For example, the cube root of 8, written as \( \sqrt[3]{8} \), asks: "What number cubed gives 8?" The answer is 2 because \( 2 \times 2 \times 2 = 8 \).

• Radicals can make expressions look simpler or more familiar.
• They allow you to express powers in a different form, which can be helpful in solving equations or simplifying complex calculations.

Understanding how to convert between radical expressions and rational exponents is a key math skill for working with these kinds of elements.

Equivalent Expressions

Equivalent expressions are different ways of writing the same mathematical quantity. This concept is crucial in algebra since being able to identify equivalent expressions can help simplify problems and make calculations easier.
For instance, \( 3x^{-1/3} \) and \( \frac{3}{\sqrt[3]{x}} \) are equivalent. Despite their different appearances, they represent the same value under any substitution of \(x\) that doesn't make the denominator zero.

• To determine equivalency, you may want to convert rational expressions into their radical forms or vice versa.
• Simplification and factoring are common methods used to recognize equivalency among expressions.

Grasping this concept means recognizing that even though expressions may be written differently, they can still represent the same numerical value or relationship.

Cube Roots

The cube root of a number is a value that, when used in three identical multiplications (cubed), gives the original number. It is the opposite of cubing a number. Symbols like \( \sqrt[3]{x} \) are used to denote cube roots, and these are particularly useful when dealing with cubic equations or simplifying certain expressions.

• Cubing a number involves multiplying it by itself twice more. For example, \( 5^3 = 5 \times 5 \times 5 = 125 \).
• Finding a cube root, like \( \sqrt[3]{125} \), involves the reverse process: asking "what number times itself twice more equals 125?" The answer is 5.

Understanding cube roots is essential for solving problems that involve cubic equations or dealing with physical quantities in three dimensions. They also play a significant role in simplifying and solving certain algebraic expressions.