Problem 69 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\frac{2}{\sqrt[3]{9}}\)

Step-by-Step Solution

Answer

The simplest radical form is \( \frac{2}{3} \sqrt[3]{3} \).

Step 1: Understand the Problem

We need to express the given expression \( \frac{2}{\sqrt[3]{9}} \) in its simplest radical form. This means dealing with the cube root in the denominator and simplifying it.

Step 2: Rationalize the Denominator

To rationalize \( \frac{2}{\sqrt[3]{9}} \), we need to eliminate the cube root from the denominator. Multiply the numerator and the denominator by \( \sqrt[3]{9^2} \) to make the denominator a cube root of a cube number.

Step 3: Multiply the Expression

Multiply the numerator and denominator by \( \sqrt[3]{81} \):\[ \frac{2 \cdot \sqrt[3]{81}}{\sqrt[3]{9} \cdot \sqrt[3]{81}} = \frac{2 \cdot \sqrt[3]{81}}{\sqrt[3]{729}}\]

Step 4: Simplify the Denominator

Since 729 is a perfect cube \((9^3)\), \(\sqrt[3]{729} = 9\). Now the expression is: \[ \frac{2 \cdot \sqrt[3]{81}}{9} \]

Step 5: Simplify the Numerator

For the numerator \( \sqrt[3]{81} \), note that 81 can be written as \( 3^4 \). Thus, \( \sqrt[3]{81} = \sqrt[3]{3^4} = 3^{4/3} = 3 \cdot \sqrt[3]{3} \).

Step 6: Combine and Simplify

Substituting back, the expression becomes: \[ \frac{2 \cdot 3 \cdot \sqrt[3]{3}}{9} \] This simplifies to: \[ \frac{6 \cdot \sqrt[3]{3}}{9} = \frac{2}{3} \cdot \sqrt[3]{3} \]

Step 7: Final Answer

The simplest radical form of the expression is \[ \frac{2}{3} \sqrt[3]{3} \].

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves removing any radical expression from the denominator of a fraction. In the expression \( \frac{2}{\sqrt[3]{9}} \), we use rationalizing as a method to handle the cube root in the denominator.
To rationalize, multiply both the numerator and the denominator by a term that can convert the radical in the denominator into a simpler form or, ideally, a whole number.
For our example, we identify that multiplying by \( \sqrt[3]{81} \) will help us convert the cubic root of 9 into a whole number because we aim for a perfect cube in the denominator.

• By multiplying, we get \( \frac{2 \cdot \sqrt[3]{81}}{\sqrt[3]{9} \cdot \sqrt[3]{81}} \).
• This results in \( \frac{2 \cdot \sqrt[3]{81}}{\sqrt[3]{729}} \), since \( 9 \cdot 81 = 729 \), which is a perfect cube.

The goal is to simplify the denominator to make calculations easier or to compare expressions more readily.

Cube Roots

Cube roots are a type of radical expression specifically dealing with numbers that need to be multiplied by themselves twice to yield the original number. If you have a number \( x \), its cube root is written as \( \sqrt[3]{x} \). It's akin to asking, "What number, when used as a factor three times, gives me \( x \)?"
In this exercise, the cube root was \( \sqrt[3]{9} \). The challenge was to simplify and rationalize this expression so that the cube root is no longer in the denominator.

• We transformed \( \sqrt[3]{9} \) by recognizing it could be modified into a perfect cube through multiplication.
• The value \( 81 \) is chosen because \( 9 \cdot 81 = 729 \), which simplifies in cubic form as \( 9^3 \).

The transformation of cube roots often makes later manipulation or calculation simpler by working with perfect cubes.

Simplifying Radicals

Simplifying radicals refers to the process of manipulating a radical expression into its most reduced form. This often involves expressing the radical in terms of smaller components that can be easily managed or simplified into a more concise expression.
In the context of this problem, after rationalizing the denominator, the next step was simplifying the expression \( \sqrt[3]{81} \).

• Recognize that \( 81 \) can be factored into \( 3^4 \), which allows breakdown into more manageable radicals.
• By converting \( \sqrt[3]{81} \) into \( 3^{4/3} \), we write it as \( 3 \cdot \sqrt[3]{3} \).

Combining these steps led us to the solution \( \frac{2}{3} \sqrt[3]{3} \), which represents the simplest radical form of the original expression. Simplification of radicals is key to creating more communicable and workable mathematical expressions in further calculations or comparisons.