Problem 19 - College Algebra with Modeling and Visualization

Question

Find the cube root of the number. $$ \frac{1}{27} $$

Step-by-Step Solution

Answer

The cube root of \( \frac{1}{27} \) is \( \frac{1}{3} \).

Step 1: Understand the Problem

We need to find the cube root of the fraction \( \frac{1}{27} \). This means we are looking for a number which, when multiplied by itself three times, will equal \( \frac{1}{27} \).

Step 2: Write the Cube Root Expression

The cube root of \( \frac{1}{27} \) can be written as \( \sqrt[3]{\frac{1}{27}} \).

Step 3: Simplify the Fraction

Notice that \( \frac{1}{27} \) can be written as \( (\frac{1}{3})^3 \) because \( 3^3 = 27 \).

Step 4: Apply the Cube Root

The property of cube roots states that \( \sqrt[3]{(a^3)} = a \). So, \( \sqrt[3]{(\frac{1}{3})^3} = \frac{1}{3} \).

Step 5: Final Answer

The cube root of \( \frac{1}{27} \) is \( \frac{1}{3} \).

Key Concepts

Rational Functions

Rational functions are a type of function represented by the quotient of two polynomials. When dealing with rational functions, it's important to understand how fractions work and how they are affected by operations like cube roots. Firstly, a rational function takes the form \( f(x) = \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomials and \( q(x) eq 0 \). This ensures the function is well-defined over its domain.

In our cube root problem, we are exploring a fraction, which can also be thought of as a basic rational function where the variable \( x \) is simply absent. When we find the cube root of a fraction, we are asking for a number which, multiplied by itself three times, results in the original fraction. This is akin to asking what function \( f(x) \) when cube rooted, gives another function or number, reinforcing the theme of rational functions and their behavior under roots.

Simplifying Fractions

Simplifying fractions is a fundamental process in mathematics, which involves reducing the fraction to its simplest form. This means expressing it as an equivalent fraction with a smaller numerator and denominator, without changing its value.

To simplify \( \frac{1}{27} \), recognize that 27 is a perfect cube, \( 27 = 3^3 \). Thus, \( \frac{1}{27} \) is really \( \left( \frac{1}{3} \right)^3 \). This insight allows us to directly apply the cube root, as taking the cube root of something cubed retrieves the base itself, being \( \frac{1}{3} \).

• First, factor the denominator if possible.
• Express the fraction as a power, if it reveals a common factor or simplifies.
• Apply root operations directly to each distinct power with confidence.

This simplification step makes the process of dealing with cube roots more intuitive and straightforward.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations combined to form expressions like \( x^2 + 3x - 4 \). While the original problem involves numerical fractions, the techniques used to solve it can be applied to algebraic expressions as well.

When you're dealing with a cube in the context of an algebraic expression, you can apply the cube root operation to variables raised to a power, just as you would with numbers. For instance, the cube root of \( x^6 \) is \( x^2 \), because \( x^2 \times x^2 \times x^2 = x^6 \).

• If the expression or its parts are polynomial, these form the basis of rational functions as explained initially.
• Simplifying involves recognizing common factors or blocks within the expression that can be separated under a root.
• The properties of exponents and roots remain uniform across numbers and variables.

Understanding algebraic expressions gives insight into transforming more complex expressions, leveraging operations like cube roots effectively.