Problem 16 - Intermediate Algebra

Question

Rationalize each denominator. See Examples 1 through 3. $$ \frac{5}{\sqrt[3]{9}} $$

Step-by-Step Solution

Answer

\(\frac{5 \cdot \sqrt[3]{3}}{3}\)

Step 1: Identify the Denominator

The denominator of the given expression is \( \sqrt[3]{9} \). The goal is to rationalize this, which involves eliminating the cube root from the denominator.

Step 2: Multiply by the Conjugate Factor

Since we are dealing with a cube root, we multiply the numerator and the denominator by the cube root of the squared denominator. This results in multiplying by \( \sqrt[3]{81} \). Thus, we multiply as follows:\[\frac{5 \cdot \sqrt[3]{81}}{\sqrt[3]{9} \cdot \sqrt[3]{81}}\]

Step 3: Simplify the Denominator

Simplifying the denominator \( \sqrt[3]{9} \cdot \sqrt[3]{81} \) gives us \( \sqrt[3]{9 \times 81} = 9 \), since cube roots follow the rule \( \sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b} \).

Step 4: Simplify the Expression

Now, we simplify the expression:\[\frac{5 \cdot \sqrt[3]{81}}{9}\]

Step 5: Final Answer

The final rationalized expression is\[\frac{5 \cdot \sqrt[3]{81}}{9}\] where \( \sqrt[3]{81} \) simplifies to \( 3 \cdot \sqrt[3]{3} \), thus:\[\frac{15 \cdot \sqrt[3]{3}}{9}\dots \text{further simplifying results in:} \frac{5 \cdot \sqrt[3]{3}}{3}\]

Key Concepts

Understanding Cube Roots

Imagine you have a number and you want to know what other number multiplied by itself twice gives you the original number. This is what a cube root does. For instance, the cube root of 8 is 2 because 2 \times 2 \times 2 equals 8. The cube root of a number is denoted by \( \sqrt[3]{a} \). Cube roots are common when working with expressions involving radicals, and when you need to rationalize a denominator with a cube root, it can be a bit more involved than dealing with square roots. To rationalize a cube root, you'd generally multiply by a factor that makes the radical a whole number. In the exercise above, we rationalized the denominator \( \sqrt[3]{9} \) by multiplying with \( \sqrt[3]{81} \) because

• it helps convert \( \sqrt[3]{9} \) into a clean number, \( 9 \).
• This process requires understanding how to factor expressions inside radicals properly.

In our case, 81 was the perfect number because when multiplied with 9, the result is a factorable cube.

Simplifying Expressions

Simplifying expressions means making them as straightforward as possible, reducing them down to the most basic terms. This often involves combining like terms, reducing fractions, and eliminating radicals when possible. In the example exercise, simplifying began by handling the denominator; \( \sqrt[3]{9} \cdot \sqrt[3]{81} \) turned into \( \sqrt[3]{9 \times 81} \), resulting in simply \( 9 \), a whole number. Then,

• The numerator \( 5 \cdot \sqrt[3]{81} \) was similarly simplified.
• The cube root 81 simplified to \( 3 \cdot \sqrt[3]{3} \) because 81 is the cube of 3 times another factor of 3.

Thus, understanding how to break down and reduce these cube roots is crucial for simplifying expressions efficiently.

Multiplying Radicals

When multiplying radicals, it's key to remember that roots can be combined under one radical if they share the same index. In simpler terms, \( \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \times b} \). This rule holds for both square roots, cube roots, and higher-index roots.By applying this rule in the problem provided, we multiplied \( \sqrt[3]{9} \) and \( \sqrt[3]{81} \) to get \( \sqrt[3]{729} \), which then simplifies directly to 9.

• Because \( 729 = 9^3 \), this works perfectly to eliminate the radical.
• The key is multiplying like radicals and simplifying them as far as possible afterward.

This method ensures that expressions are not just rationalized, but also kept clean and simple. Mastering the art of multiplying radicals is essential for both simplifying and rationalizing complex expressions.