Problem 21 - Intermediate Algebra

Question

Find each cube root. $$ \sqrt[3]{\frac{1}{8}} $$

Step-by-Step Solution

Answer

The cube root of \( \frac{1}{8} \) is \( \frac{1}{2} \).

Step 1: Identify the Problem

We need to find the cube root of \( \frac{1}{8} \). A cube root of a number \( x \) is a number \( y \) such that \( y^3 = x \).

Step 2: Express as Cube Root Equation

In mathematical terms, the problem can be expressed as \( y = \sqrt[3]{\frac{1}{8}} \), which implies that \( y^3 = \frac{1}{8} \).

Step 3: Find Cube Root of the Numerator and Denominator

Since \( \frac{1}{8} \) is a fraction, we find the cube root of the numerator and the denominator separately. The cube root of 1 is 1 because \( 1^3 = 1 \).

Step 4: Calculate the Cube Root of the Denominator

The cube root of 8 is 2 because \( 2^3 = 8 \).

Step 5: Write the Result as a Fraction

Thus, the cube root of \( \frac{1}{8} \) is \( \frac{1}{2} \). This is because \( \left( \frac{1}{2} \right)^3 = \frac{1^3}{2^3} = \frac{1}{8} \).

Key Concepts

Understanding Fractions

Fractions are a way to represent a part of a whole. They consist of two numbers: the numerator and the denominator. The numerator is the top number and indicates how many parts we have. The denominator is the bottom number and tells us how many equal parts the whole is divided into.
Understanding fractions is crucial in math because they allow us to express numbers that are not whole. For example, the fraction \(\frac{1}{8}\) means we have one part out of a total of eight equal parts.
It's important to grasp how fractions work because they're used in many mathematical operations, like adding, subtracting, multiplying, or finding roots, as in our exercise.

Numerator and Denominator Roles

Let's take a closer look at what the numerator and denominator do. In a fraction like \(\frac{1}{8}\), the numerator (1) indicates that we have one part. The denominator (8) shows the size of one whole divided into eight equal parts.
The numerator and denominator work together to describe the value of the fraction. You can think of the denominator as the "parts maker," and the numerator tells you how many of those parts you actually have.
In operations like finding the cube root, understanding these roles is crucial because we might need to handle the numerator and denominator separately. This brings us to solving cube root equations.

Solving Cube Root Equations

A cube root equation asks us to find a number that, when multiplied by itself three times, results in the given number. For instance, finding the cube root of \(\frac{1}{8}\) means discovering a number that satisfies:

• \(y^3 = \frac{1}{8}\)

To do this, we have two main actions:
1.

• Find the cube root of the numerator.
• Find the cube root of the denominator.

Once we know these roots, we can express the fraction's cube root.
Understanding this process allows us to tackle more complex equations while keeping our work organized and systematic.

Step by Step Solution Process

Breaking down solutions into steps helps us tackle problems methodically. Let's revisit the solution to \(\sqrt[3]{\frac{1}{8}}\) step by step:
1. Identify the problem and express it in math terms as a cube root equation: \(y = \sqrt[3]{\frac{1}{8}}\).
2. Recognize that the fraction's cube root involves separate calculations for the numerator and denominator.
3. Calculate the cube root of 1, which is 1 because \(1^3 = 1\), and the cube root of 8, which is 2 because \(2^3 = 8\).
4. Put these two results together to form the fraction \(\frac{1}{2}\), ensuring that \(\left(\frac{1}{2}\right)^3 = \frac{1}{8}\).

• This careful, step-by-step approach ensures accuracy and deepens our understanding of math operations like cube roots.