Problem 7 - 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

Option C: \(\frac{1}{\sqrt[3]{3 x}}\)

Step 1: Identify the Rational Exponent

Examine the expression to identify the rational exponent. The given expression is \((3x)^{-\frac{1}{3}}\). Here, \(3x\) is the base and \(-\frac{1}{3}\) is the exponent.

Step 2: Understand the Meaning of the Exponent

The exponent of \(-\frac{1}{3}\) indicates taking the cube root of the reciprocal of the base. The negative sign means you will take the reciprocal of the expression, and \(\frac{1}{3}\) means taking the cube root. So, it implies \(\frac{1}{\sqrt[3]{3x}}\).

Step 3: Match with the Given Options

Look for the option that represents \(\frac{1}{\sqrt[3]{3x}}\). Comparing options A-H, you find that option C, which is \(\frac{1}{\sqrt[3]{3x}}\), matches the expression you've derived.

Key Concepts

Radical Expressions

Radical expressions involve roots, like square roots or cube roots. They are alternative ways to express numbers or variables under a root symbol. It's essential to understand these, as they allow simplifying or rewriting expressions involving exponents.

• The most common radical is the square root, denoted by the symbol \( \sqrt{} \).
• Cube roots are another type. Here, you find a number that, when multiplied by itself three times, returns the original number. It's shown as \( \sqrt[3]{} \).
• Radicals can transform into expressions with fractional exponents and vice versa. This relationship helps in solving equations and simplifying algebraic expressions.

For instance, \( x^{1/3} \) is equivalent to \( \sqrt[3]{x} \). The conversion between these forms allows versatility in problem-solving.

Reciprocal

Reciprocals are fundamental in mathematics, particularly when dealing with fractions and rational exponents. The reciprocal of a number is 1 divided by that number.

• If you have a positive number \( a \), its reciprocal is \( \frac{1}{a} \).
• In terms of exponents, a negative exponent, like \( x^{-n} \), implies the reciprocal of the base raised to the positive EXponent: \( \frac{1}{x^n} \).
• This concept is crucial when transforming expressions like \( (3x)^{-1/3} \) into \( \frac{1}{(3x)^{1/3}} \).

Understanding reciprocals helps when rewriting and simplifying expressions with negative powers, making complex problems much easier to manage.

Cube Root

The cube root of a number is a value that, when multiplied by itself three times, results in the original number. This concept is vital in solving equations where the variable is raised to the third power.

• Cube roots are represented as \( \sqrt[3]{x} \).
• They serve as inverses to cubes. If \( y^3 = x \), then \( y = \sqrt[3]{x} \).
• When dealing with rational exponents, a cube root corresponds to an exponent of \( \frac{1}{3} \).

In our example, \( (3x)^{-1/3} \), taking the cube root of \( 3x \) and considering the reciprocal yields \( \frac{1}{\sqrt[3]{3x}} \), matching option C. This importance of cube roots is evident in transforming and simplifying expressions in mathematics.