Problem 27 - Algebra 2 and Trigonometry

Question

In \(11-38,\) evaluate each expression in the set of real numbers. $$ \sqrt[3]{\frac{8}{27}} $$

Step-by-Step Solution

Answer

The value of \( \sqrt[3]{\frac{8}{27}} \) is \( \frac{2}{3} \).

Step 1: Understand the Problem

We are tasked with evaluating the expression \( \sqrt[3]{\frac{8}{27}} \). This implies finding the cube root of the fraction \( \frac{8}{27} \).

Step 2: Evaluate the Cube Root of the Fraction

To find \( \sqrt[3]{\frac{8}{27}} \), we take the cube root of the numerator and the denominator separately. That is, \( \sqrt[3]{8} \) and \( \sqrt[3]{27} \).

Step 3: Calculate Each Cube Root Individually

\( \sqrt[3]{8} = 2 \) because \( 2 \times 2 \times 2 = 8 \). Similarly, \( \sqrt[3]{27} = 3 \) because \( 3 \times 3 \times 3 = 27 \).

Step 4: Combine the Results

Combine the results from Step 3 to find \( \sqrt[3]{\frac{8}{27}} = \frac{\sqrt[3]{8}}{\sqrt[3]{27}} = \frac{2}{3} \).

Key Concepts

Evaluating Expressions

Evaluating expressions, especially those involving cube roots, is a foundational skill in mathematics. When we talk about evaluating an expression, we mean simplifying it down to its most basic form or finding its value. For this task, we evaluated an expression that asked us to find the cube root.

To evaluate \[ \sqrt[3]{\frac{8}{27}} \], we need to break it down into simpler parts.

• First, recognize that the expression involves a fraction within the cube root symbol.
• Second, understand that finding the cube root of a fraction involves finding the cube roots of both the numerator and the denominator separately.

This breakdown method allows us to manage complex expressions more effectively. Once broken down, the calculations become straightforward. Also, evaluating expressions often requires knowledge of different mathematical functions and operations, such as recognizing common roots and powers.

Once the individual cube roots in the numerator and denominator are evaluated, they can be combined to give the final simplified fraction.

Real Numbers

The set of real numbers is vast and includes a variety of numbers that we work with in mathematics. Real numbers encompass:

• Natural numbers
• Whole numbers
• Integers
• Rational numbers (like fractions and decimals)
• Irrational numbers (like \( \pi \) or the square root of non-perfect squares)

These numbers can be ordered on a number line, giving us a way to visually represent and compare them.

In our exercise, both 8 and 27 are part of the real numbers. The cube root of a real number is always a real number. Here, when we found \[ \sqrt[3]{8} = 2 \] and \[ \sqrt[3]{27} = 3 \], the results, 2 and 3, are clearly real numbers.

Understanding real numbers is vital for solving expressions, as it ensures we interpret the numbers within their correct context and constraints.

Fractions

Fractions are a way to represent parts of a whole, using two numbers: a numerator and a denominator. The top part is the numerator, and the bottom part is the denominator. Fractions are a key component of rational numbers, which are a subset of real numbers.

In our evaluation task, \[ \sqrt[3]{\frac{8}{27}} \]involves the fraction \[ \frac{8}{27}. \] The process of finding cube roots for fractional numbers involves handling the numerator and the denominator separately. This can be done without combining them initially.

For instance, we solved \[ \sqrt[3]{8} = 2 \] (and similarly for the denominator) to get a simple fraction.

• The cube root of the numerator 8 resulted in 2.
• The cube root of the denominator 27 resulted in 3.

Thus, our initial fractional expression simplified into \[ \frac{2}{3}. \] This result shows the importance of understanding fractions not just as static entities, but as flexible numbers that can interact with operations like finding roots.