Problem 63 - Precalculus: Functions and Graphs

Question

Rationalize the denominator. $$\frac{1}{\sqrt[3]{a}-\sqrt[3]{b}}$$

Step-by-Step Solution

Answer

The rationalized form is \( \frac{(\sqrt[3]{a})^2 + \sqrt[3]{a}\sqrt[3]{b} + (\sqrt[3]{b})^2}{a - b} \).

Step 1: Understanding the Problem

We need to rationalize the denominator of the expression \( \frac{1}{\sqrt[3]{a} - \sqrt[3]{b}} \). This means we need to eliminate the cube root from the denominator.

Step 2: Identify the Conjugate

To rationalize a denominator with a form \( \sqrt[3]{a} - \sqrt[3]{b} \), we use its conjugate, \( (\sqrt[3]{a})^2 + \sqrt[3]{a}\sqrt[3]{b} + (\sqrt[3]{b})^2 \). This is derived from factoring a difference of cubes.

Step 3: Multiply by the Conjugate

Multiply the numerator and the denominator by the conjugate: \(\frac{1}{\sqrt[3]{a} - \sqrt[3]{b}} \cdot \frac{(\sqrt[3]{a})^2 + \sqrt[3]{a}\sqrt[3]{b} + (\sqrt[3]{b})^2}{(\sqrt[3]{a})^2 + \sqrt[3]{a}\sqrt[3]{b} + (\sqrt[3]{b})^2} \).

Step 4: Simplify the Denominator

The denominator simplifies as follows, using the identity for the difference of cubes: \((\sqrt[3]{a})^3 - (\sqrt[3]{b})^3 = a - b\). Thus, the denominator becomes \((\sqrt[3]{a})^3 - (\sqrt[3]{b})^3 = a - b\).

Step 5: Write the Rationalized Expression

With the denominator now \(a - b\), the full expression becomes: \(\frac{(\sqrt[3]{a})^2 + \sqrt[3]{a}\sqrt[3]{b} + (\sqrt[3]{b})^2}{a - b}\). This is the rationalized form of the original expression.

Key Concepts

Cube Roots

Understanding cube roots is crucial for rationalizing expressions with roots in the denominator. A cube root, represented as \( \sqrt[3]{x} \), is a number that, when multiplied by itself three times, results in \( x \). Similar to square roots, cube roots help in simplifying complex algebraic expressions.

• For example, the cube root of 27 is 3 because \( 3 \times 3 \times 3 = 27 \).
• Cube roots can be positive or negative, depending on the sign of \( x \). For instance, the cube root of -27 is -3.

Recognizing these roots helps in manipulating and simplifying expressions through operations like multiplication or finding conjugates.

Conjugates in Algebra

Conjugates are a pair of expressions that, when multiplied together, eliminate certain terms such as radicals. Using conjugates in algebra, especially with roots, is a powerful technique to simplify expressions and rationalize denominators.

• The conjugate of \( \sqrt[3]{a} - \sqrt[3]{b} \) is \((\sqrt[3]{a})^2 + \sqrt[3]{a}\sqrt[3]{b} + (\sqrt[3]{b})^2 \).
• This works by applying the identity for the difference of cubes, removing cube roots from the denominator.

Understanding and using conjugates in algebra allows for the simplification of expressions and ensures that all radicals are represented in the numerator only, making calculations more manageable.

Difference of Cubes

The difference of cubes is a special algebraic identity that helps in rationalizing denominators with cube roots. It states that \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \).

• This identity means that when you have a cube root difference in a denominator, you can multiply by the conjugate to simplify.
• For example, in the expression \( \frac{1}{\sqrt[3]{a} - \sqrt[3]{b}} \), multiplying by the conjugate yields the denominator as \( a - b \).

Applying the difference of cubes allows us to transform expressions into a simpler form, free of radicals in the denominator, facilitating easier integration or differentiation in further mathematical processes.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator consist of polynomial expressions. They can often include roots, like cube roots, which need to be simplified for the expression to be rationalized.

• The goal of rationalizing such expressions is to get rid of radicals from the denominator for easier computation.
• For example, rationalizing \( \frac{1}{\sqrt[3]{a} - \sqrt[3]{b}} \) by multiplying by its conjugate simplifies it to have a polynomial denominator \( a - b \).

Understanding how to manipulate these expressions through operations such as finding conjugates or using algebraic identities ensures that you can efficiently process and solve rational expressions in algebraic contexts.