Problem 75 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The cube root of \( \frac{8}{27} \) simplifies to \( \frac{2}{3} \).

Step 1: Recognize the Problem

You need to simplify the cube root of a fraction, which means you need to simplify \( \sqrt[3]{\frac{8}{27}} \).

Step 2: Break Down the Fraction

Identify the cube root separately in the numerator and the denominator of the fraction, \( \sqrt[3]{8} \) and \( \sqrt[3]{27} \).

Step 3: Simplify the Numerator

Recognize that 8 is a perfect cube, which can be simplified as \( 8 = 2^3 \). Thus, the cube root \( \sqrt[3]{8} = 2 \).

Step 4: Simplify the Denominator

Recognize that 27 is also a perfect cube, which can be simplified as \( 27 = 3^3 \). Thus, the cube root \( \sqrt[3]{27} = 3 \).

Step 5: Combine the Results

Combine the simplified components: \( \frac{2}{3} \). Hence, \( \sqrt[3]{\frac{8}{27}} = \frac{2}{3} \).

Key Concepts

Fractional Roots

Understanding fractional roots is essential when dealing with expressions like \( \sqrt[3]{\frac{8}{27}} \). A fractional root signifies taking the root of both the numerator and the denominator separately. In the case of a cube root, you are looking for a number which, when multiplied by itself three times, gives the original number, both in the numerator and the denominator.

When simplifying fractional roots:

• Separate the expression into two distinct parts: the numerator and the denominator.
• Apply the cube root to each part individually.
• This process helps simplify the overall expression.

For the expression \( \sqrt[3]{\frac{8}{27}} \):

• The cube root of 8 is 2.
• The cube root of 27 is 3.
• Thus, the expression simplifies to \( \frac{2}{3} \).

Breaking down such fractions into smaller, manageable parts makes them much easier to handle. This step-by-step simplification process is fundamental in mastering fractional roots.

Perfect Cubes

Perfect cubes are numbers that can be expressed as the cube of an integer. For example, 8 is a perfect cube because \( 2^3 = 8 \). Similarly, 27 is a perfect cube because \( 3^3 = 27 \). Recognizing perfect cubes is crucial when simplifying cube roots because it allows for direct simplification.

To determine if a number is a perfect cube:

• Find its prime factors.
• See if these factors can be grouped into triples.

Let's consider some common perfect cubes:

• 1: \( 1^3 \)
• 8: \( 2^3 \)
• 27: \( 3^3 \)
• 64: \( 4^3 \)
• 125: \( 5^3 \)

Recognizing these helps in simplifying expressions quickly. When simplifying \( \sqrt[3]{8} \) and \( \sqrt[3]{27} \), knowing they are perfect cubes makes it straightforward, directly resulting in the integers 2 and 3 respectively.

Simplifying Radicals

Simplifying radicals involves reducing the complexity of expressions involving roots. The main goal is to express them in their simplest form, which often requires recognizing patterns or known values (like perfect cubes or squares). With cube roots, simplify the expression by identifying and utilizing perfect cubes.

Here are the steps to simplify a radical, specifically a cube root:

• Identify the cube roots of both the numerator and the denominator separately in fractional expressions.
• Simplify each part by finding the cube root of any perfect cubes.
• Combine these results to obtain a simplified fraction.

In the expression \( \sqrt[3]{\frac{8}{27}} \), you simplify it by recognizing both 8 and 27 as perfect cubes, yielding \( \frac{2}{3} \).

Being able to break down and simplify radicals is a key skill in algebra. It not only helps in handling various mathematical problems but also allows for deeper understanding and manipulation of complex numerical expressions.