Problem 72 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\frac{\sqrt[3]{8}}{\sqrt[3]{16}}\)

Step-by-Step Solution

Answer

The simplest radical form is \( \frac{\sqrt[3]{4}}{2} \).

Step 1: Understand the Problem

We need to simplify the expression that contains cube roots. The expression given is \( \frac{\sqrt[3]{8}}{\sqrt[3]{16}} \). We aim to express both the numerator and the denominator in their simplest radical form.

Step 2: Simplify the Numerator

Start by simplifying \( \sqrt[3]{8} \). Since 8 can be written as \( 2^3 \), the cube root of 8 is \(2\). Thus, \( \sqrt[3]{8} = 2 \).

Step 3: Simplify the Denominator

Now simplify \( \sqrt[3]{16} \). Since 16 can be written as \( 2^4 \), we can express it as \( 2^3 \times 2^1 \). The cube root of \(2^3\) is \(2\), so, \( \sqrt[3]{16} = 2\sqrt[3]{2} \).

Step 4: Divide the Simplified Numerator by the Simplified Denominator

Now that we have simplified the numerator to 2 and the denominator to \(2\sqrt[3]{2}\), our expression becomes: \( \frac{2}{2\sqrt[3]{2}} \). Simplify this further by canceling the 2 in the numerator and denominator, yielding \( \frac{1}{\sqrt[3]{2}} \).

Step 5: Express the Answer in Simplest Radical Form

To express \( \frac{1}{\sqrt[3]{2}} \) in radical form without a radical in the denominator, multiply the numerator and denominator by \(\sqrt[3]{4}\). This gives us \( \frac{\sqrt[3]{4}}{\sqrt[3]{8}} = \frac{\sqrt[3]{4}}{2} \). Thus, the expression in simplest radical form is \( \frac{\sqrt[3]{4}}{2} \).

Key Concepts

Cube Roots

A cube root is a number that, when multiplied by itself two more times (total three times), results in the original number. For example, the cube root of 8 is 2, because when you multiply 2 \( \times 2 \times 2 \) (which is \(2^3\)), the outcome is 8. Similarly, the cube root of 27 is 3, as \(3^3\) equals 27.
Understanding cube roots can simplify expressions containing radicals by converting complex numbers into more manageable small numbers.

• Recognizing powers: For cube roots, look for numbers that can be expressed as a base raised to the power of three (like \(2^3\)).
• Extraction: Apply the cube root to simplify the expression (i.e., \(\sqrt[3]{27} = 3\)).

In expressions like \(\frac{\sqrt[3]{8}}{\sqrt[3]{16}}\), simplifying both the numerator and denominator using their cube roots makes it easier to work with and understand.

Simplest Radical Form

The simplest radical form of a number is when the expression of that number under a radical (or root) is made as simplified as possible. This means eliminating any possible radicals in the denominator or breaking down numbers into their lowest factors.
When simplifying radicals, the goal is to make use of the properties of exponents and roots effectively.

• Factorization: Break down the number under the radical to its prime factors. For instance, 16 can be written as \( 2^4 \).
• Root Extraction: Simplify by extracting roots where possible. For \( \sqrt[3]{8} \), this simplifies to 2, while \(\sqrt[3]{16}\) becomes \( 2\sqrt[3]{2} \).

By expressing both the numerator and the denominator in their simplest radical form, you are crafting a more understandable expression that makes further calculations easier, as seen in the solution.

Rationalizing the Denominator

Rationalizing the denominator is the process of eliminating radicals (like \(\sqrt{2}\) or \(\sqrt[3]{2}\)) from the denominator of a fraction. This is often done to simplify the appearance of a mathematical expression and adhere to a standard form.
In the exercise, the fraction \(\frac{1}{\sqrt[3]{2}}\) is not considered fully simplified due to the radical in the denominator.

• Multiplication Trick: Multiply both the numerator and denominator by the radical necessary to clear the denominator. For example, multiply \(\sqrt[3]{2}\) by \(\sqrt[3]{4}\) to get a rational denominator.
• Final Expression: After rationalizing, \(\frac{1}{\sqrt[3]{2}}\) becomes \(\frac{\sqrt[3]{4}}{2}\) because \(\sqrt[3]{2} \times \sqrt[3]{4} = \sqrt[3]{8} = 2\).

Overall, rationalizing the denominator helps to keep mathematical expressions neat and more easily understood in terms of further calculations or simplifications.