Problem 6 - Algebra A Combined Function

Question

Rationalize each denominator. Assume that all variables represent positive real numbers. \(\frac{6}{\sqrt[3]{9}}\)

Step-by-Step Solution

Answer

Rationalized as \( \frac{2 \sqrt[3]{81}}{3} \).

Step 1: Identify the Denominator

The denominator in the expression \( \frac{6}{\sqrt[3]{9}} \) is \( \sqrt[3]{9} \). We need to rationalize this expression.

Step 2: Understand Rationalization

Rationalization involves removing the radical from the denominator. For cube roots like \( \sqrt[3]{9} \), we multiply by appropriate terms to make the denominator a perfect cube.

Step 3: Determine Correct Multiplier

To rationalize, we will multiply by \( \sqrt[3]{9^2} \) such that \( \sqrt[3]{9} \times \sqrt[3]{9^2} = \sqrt[3]{729} = 9 \), which is a rational number.

Step 4: Multiply Numerator and Denominator

Multiply both the numerator and the denominator by \( \sqrt[3]{9^2} \):\[ \frac{6}{\sqrt[3]{9}} \times \frac{\sqrt[3]{9^2}}{\sqrt[3]{9^2}} = \frac{6 \sqrt[3]{81}}{9} \]

Step 5: Simplify the Expression

Simplify the resulting expression: \[ \frac{6 \sqrt[3]{81}}{9} = \frac{2 \sqrt[3]{81}}{3} \]

Key Concepts

Cube Roots

The concept of cube roots involves identifying a number that, when multiplied by itself twice, gives the original number under the cube root. For instance, the cube root of 8 is 2 since when you multiply 2 by itself twice (i.e., \(2 \times 2 \times 2\)), the product is 8. Cube roots are represented using the symbol \(\sqrt[3]{\cdot}\) and operate similarly to square roots but pertain to cubing instead.
Understanding cube roots is essential when dealing with problems that involve radicals in the denominator, as it allows for simplification and rationalization of expressions.

Radicals

Radicals refer to expressions that contain a root symbol, such as square roots, cube roots, and higher-order roots. The term under the root is referred to as the radicand. For example, \(\sqrt[3]{9}\) is a radical where 9 is the radicand, and 3 indicates that it's a cube root.
Radicals can often make mathematical expressions complex, especially when they are in the denominator. Simplifying or rationalizing radicals is a method to change them into a more manageable form. This aids in solving equations or evaluating expressions with more ease.
When rationalizing, our goal is to eliminate the root from the denominator so that the expression becomes easier to handle and free from radicals at the bottom.

Rational Numbers

Rational numbers are any numbers that can be expressed as the quotient or fraction \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b eq 0\). These numbers include integers, fractions, and terminating or repeating decimals.
A key aspect of rational numbers is that they do not include any roots unless they simplify completely. For instance, \(\sqrt{4}\) simplifies to 2, which is rational, whereas \(\sqrt{3}\) remains irrational.
Understanding rational numbers is crucial when simplifying expressions because we aim to convert irrational numbers into rational ones when rationalizing denominators.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form while maintaining equality. It often requires combining like terms, reducing fractions, or eliminating radicals.
In the exercise example, to simplify \(\frac{6 \sqrt[3]{81}}{9}\), we divide both the numerator and the denominator by 3, which leads to \(\frac{2 \sqrt[3]{81}}{3}\).
The simplified expression is often easier to interpret or further calculate, especially when dealing with longer and more complex algebraic problems. Simplified expressions provide a clearer view of the problem's structure, making it easier to solve.