Problem 61 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 57-80: Simplify the expression, and rationalize the denominator when appropriate. $$ \frac{1}{\sqrt[3]{2}} $$

Step-by-Step Solution

Answer

\( \frac{\sqrt[3]{4}}{2} \)

Step 1: Understanding the problem

We have an expression \( \frac{1}{\sqrt[3]{2}} \). Our task is to simplify it and rationalize the denominator. Rationalizing the denominator means eliminating any roots in the denominator.

Step 2: Identify the need for rationalization

The denominator of the fraction \( \frac{1}{\sqrt[3]{2}} \) is \( \sqrt[3]{2} \), which is a cube root. We need to get rid of this cube root to rationalize.

Step 3: Rationalize the denominator

To rationalize \( \frac{1}{\sqrt[3]{2}} \), multiply both the numerator and denominator by \( \sqrt[3]{4} \), because \( \sqrt[3]{2} \times \sqrt[3]{4} = \sqrt[3]{8} = 2 \). Thus, the expression becomes \( \frac{\sqrt[3]{4}}{2} \).

Step 4: Simplify the expression

Now that the denominator is rationalized, the expression \( \frac{\sqrt[3]{4}}{2} \) is in its simplest form.

Key Concepts

Cube Roots

When we talk about cube roots, we are referring to the number which, when multiplied by itself three times, results in a given number. For instance, the cube root of 8 is 2 because 2 multiplied by itself twice (2x2x2) equals 8. The cube root is represented usually by the radical sign with an index of three, like this: \( \sqrt[3]{ } \).
Cube roots differ from square roots as they involve three iterations instead of two. These are important in algebra because they allow us to express quantities that have been cubed, now in their original form.
Often in algebra, we need to manipulate these cube roots to simplify expressions and to rationalize denominators. This process might involve multiplying by other roots or fractions to create whole numbers or a more straightforward form.

Simplifying Expressions

Simplifying expressions is an essential skill in mathematics, allowing us to make complex equations easier to work with by reducing them to their simplest form. This process involves reducing the number of operations and removing any unnecessary components.
To simplify an expression with a radical, such as cube roots, you often need to find ways to combine or eliminate the radical. Simplifying might include:

• Combining like terms - Usually reducing fractions or adding/subtracting common terms.
• Multiplying and factoring - Often used to remove radicals from denominators, particularly when rationalizing.
• Rationalizing the denominator - This means making the denominator a rational number by eliminating the roots.

In the context of the given equation, \( \frac{1}{\sqrt[3]{2}} \), simplifying involved multiplying both the numerator and denominator by a cubed version to remove the root from the denominator, resulting in a simpler form: \( \frac{\sqrt[3]{4}}{2} \).

Algebraic Fractions

Algebraic fractions are fractions where the numerator, denominator, or both are algebraic expressions, such as polynomials or radicals like cube roots. Dealing with algebraic fractions often involves simplifying and rationalizing to make equations easier to solve or understand.
Rationalizing the denominator is a typical process used when your algebraic fraction has a radical in the denominator, as seen in the problem \( \frac{1}{\sqrt[3]{2}} \).
Steps typically involve:

• Identifying if the denominator is irrational and needs rationalization.
• Multiplying the fraction by a form of 1 - a fraction which will eliminate the radical. This involves the radical multiplied in such a way that results in a whole number.
• Simplifying the resulting expression further if necessary.

By following these steps, algebraic fractions become more manageable and are expressed in a more easily understood form.