Problem 75 - Intermediate Algebra

Question

Simplify each cube root. See Example 6. $$ \sqrt[3]{\frac{8}{27}} $$

Step-by-Step Solution

Answer

\( \sqrt[3]{\frac{8}{27}} = \frac{2}{3} \).

Step 1: Understanding the Problem

The problem asks us to simplify the cube root of a fraction: \( \sqrt[3]{\frac{8}{27}} \). This means we need to find the cube root of the numerator and the denominator individually, and then simplify the fraction if possible.

Step 2: Finding the Cube Roots

First, we identify the cube roots. We note that \( 8 \) can be expressed as \( 2^3 \) because \( 2 \times 2 \times 2 = 8 \), and \( 27 \) can be expressed as \( 3^3 \) since \( 3 \times 3 \times 3 = 27 \). This means \( \sqrt[3]{8} = 2 \) and \( \sqrt[3]{27} = 3 \).

Step 3: Simplifying the Expression

Using the results from Step 2, we substitute back into the original expression: \( \sqrt[3]{\frac{8}{27}} = \frac{\sqrt[3]{8}}{\sqrt[3]{27}} = \frac{2}{3} \).

Step 4: Final Simplification Check

We check if the resulting fraction \( \frac{2}{3} \) can be simplified further. Since 2 and 3 have no common factors other than 1, this is already in its simplest form.

Key Concepts

Fraction Simplification

Simplifying fractions is at the heart of many math problems. It involves reducing a fraction to its simplest form. Simplification is achieved by finding a common factor between the numerator and the denominator.

For example, if you have a fraction like \( \frac{8}{12} \), you can simplify it. First, identify the common factors of the numerator (8) and the denominator (12). The common factor here is 4.

Divide both the numerator and the denominator by 4 to get \( \frac{2}{3} \). This is more simplified.

Always remember, a fraction is considered fully simplified when there are no common factors left between the numerator and the denominator other than 1.

• Identify common factors.
• Divide numerator and denominator by greatest common factor (GCF).
• Ensure the result cannot be simplified further.

Numerator and Denominator Cube Roots

Finding the cube roots of both the numerator and the denominator separately is a crucial step in fraction cube root problems. It simplifies the expression dramatically.

Let's take the example of finding the cube root of the fraction \( \frac{8}{27} \). The goal is to find cube roots separately for 8 and 27.

To do this, recognize the perfect cubes within these numbers:

• 8 is equivalent to \( 2^3 \), so \( \sqrt[3]{8} = 2 \).
• 27 is equivalent to \( 3^3 \), so \( \sqrt[3]{27} = 3 \).

Now you're left with a simplified fraction \( \frac{2}{3} \). Understanding this principle can help you tackle similar problems with ease.

Simplifying Rational Expressions

A rational expression is a fraction involving polynomials. Simplifying these kinds of fractions uses similar principles as numeric fraction simplification, but may involve algebraic techniques.

To simplify any rational expression, follow these steps:

• Factor the numerator and the denominator completely.
• Identify common factors in both the numerator and denominator.
• Cancel out these common factors from both sides, ensuring that the expression remains valid.

This process often involves recognizing specific patterns like difference of squares or perfect cubes, just like in numeric expressions.

After canceling, it's crucial to recheck each component of the rational expression to ensure it cannot be simplified further. This leads to the most simplified version of the rational expression, optimizing your calculations.