Problem 14 - Algebra A Combined Function

Question

Rationalize each denominator. Assume that all variables represent positive real numbers. \(\frac{5}{\sqrt[3]{9}}\)

Step-by-Step Solution

Answer

The rationalized expression is \( \frac{5 \sqrt[3]{81}}{9} \).

Step 1: Understanding the Problem

We need to rationalize the denominator of the expression \( \frac{5}{\sqrt[3]{9}} \). This means we'll eliminate the radical in the denominator by making it into a rational number.

Step 2: Identify the Cubic Root

The denominator is \( \sqrt[3]{9} \). Our goal is to eliminate the cube root from the denominator. We can do this by multiplying both the numerator and the denominator by an appropriate expression that will make the denominator a perfect cube.

Step 3: Find the Appropriate Multiplier

To make \( \sqrt[3]{9} \) a perfect cube, we multiply it by \( (\sqrt[3]{9})^2 \). Thus, we multiply the numerator and the denominator by \( \sqrt[3]{9}^2 \). This gives us: \[ \frac{5 \times \sqrt[3]{9}^2}{\sqrt[3]{9} \times \sqrt[3]{9}^2} \].

Step 4: Simplify the Denominator

Since \( \sqrt[3]{9} \times \sqrt[3]{9}^2 = (\sqrt[3]{9})^3 = 9 \), the new denominator becomes 9.

Step 5: Simplify the Numerator

The numerator becomes \( 5 \times \sqrt[3]{9}^2 \). This simplifies to \( 5 \sqrt[3]{81} \).

Step 6: Rewrite the Expression

The expression can now be rewritten as \( \frac{5 \sqrt[3]{81}}{9} \). The denominator is rationalized, and the expression is simplified.

Step 7: Final Simplified Expression

Thus, the rationalized expression is \( \frac{5 \sqrt[3]{81}}{9} \), where the denominator, 9, is a rational number.

Key Concepts

Cube Roots

Cube roots help us find a number that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2, because

• 2 × 2 × 2 = 8.

Understanding cube roots is crucial when dealing with expressions like \( \frac{5}{\sqrt[3]{9}} \).
When we say \( \sqrt[3]{9} \), we are asking, "What number cubed gives us 9?"
To rationalize such expressions, we must focus on eliminating the cube root from the denominator. This paves the way for easier computation and simplifies the expression.
Cube roots often pop up when simplifying radicals and rationalizing denominators, making it a vital concept in algebra.
Mastering cube roots will enable you to tackle a range of math problems with efficiency and clarity.

Perfect Cubes

Perfect cubes are numbers that can be expressed as another entire number raised to the power of three. Examples of perfect cubes include

• 8 (because 2³ = 8)
• 27 (because 3³ = 27)
• 64 (because 4³ = 64)

Perfect cubes are important because they help us rationalize the denominators involving cube roots.
A cube root can be easily simplified when it is a perfect cube.
In the expression \( \frac{5}{\sqrt[3]{9}} \), [9] needed to be converted into a perfect cube to remove the cube root from the denominator.
This was achieved by multiplying the expression by \((\sqrt[3]{9})^2\), which, when combined with \(\sqrt[3]{9}\), results in 9, a rational number.
Recognizing and transforming numbers into perfect cubes facilitates the simplification of complex expressions.

Simplifying Radicals

Simplifying radicals involves reducing complex root expressions to simpler forms. For cube roots, this often requires transforming the expression to ensure the radical is eliminated from the denominator.
The primary goal of simplification is to make calculations more straightforward and readable.
For instance, in the task of rationalizing \(\frac{5}{\sqrt[3]{9}}\), it was necessary to make the denominator a rational number.
To do this, we multiplied both parts of the expression by \((\sqrt[3]{9})^2\), resulting in a simplified expression \(\frac{5 \sqrt[3]{81}}{9}\).
This step was crucial for simplifying the radical in the denominator and making the whole expression clearer.
Remember, fewer radicals mean simpler calculations, which is why simplifying radicals should always be your end goal.