Problem 20 - Intermediate Algebra

Question

Use the quotient rule to simplify. See Examples 2 and 3 . $$ \sqrt[3]{\frac{3}{64}} $$

Step-by-Step Solution

Answer

\( \frac{\sqrt[3]{3}}{4} \)

Step 1: Understand the Cube Root

The expression \( \sqrt[3]{\frac{3}{64}} \) indicates that we need to find the cube root of the fraction \( \frac{3}{64} \). The cube root of a fraction \( \frac{a}{b} \) is the same as \( \frac{\sqrt[3]{a}}{\sqrt[3]{b}} \).

Step 2: Simplify the Denominator

Find the cube root of \( 64 \). Since \( 64 = 4^3 \), we have \( \sqrt[3]{64} = 4 \).

Step 3: Simplify the Numerator

The cube root of \( 3 \) is \( \sqrt[3]{3} \). Since \( 3 \) is not a perfect cube, \( \sqrt[3]{3} \) remains as is.

Step 4: Write the Simplified Expression

Now, the expression becomes: \( \frac{\sqrt[3]{3}}{4} \). This is the simplest form of the original expression.

Key Concepts

Cube Root

The concept of the cube root revolves around finding a number which, when multiplied by itself twice, yields the original number. The cube root is indicated by the symbol \( \sqrt[3]{\ } \). To better visualize this concept, let's consider the cube root of a fraction, for example, \( \sqrt[3]{\frac{3}{64}} \). Here, we need to determine the cube root of both the numerator and the denominator separately.

• Numerator: The cube root of 3, \( \sqrt[3]{3} \), remains unchanged because 3 isn't a perfect cube.
• Denominator: Now, for 64, since 64 is 4 cubed (i.e., \( 4^3 = 64 \)), the cube root becomes simply 4. Thus \( \sqrt[3]{64} = 4 \).

By addressing these separately, we can accurately find the cube root of the entire fraction.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra, allowing us to make complex mathematical expressions easier to handle. When dealing with cube roots, the goal is to break down the expression into the simplest form.

In our example, \( \sqrt[3]{\frac{3}{64}} \), we aim to simplify by taking cube roots of both the numerator and denominator. This technique allows us to transform a potentially cumbersome expression into something more digestible.

When simplifying, remember:

• Check for perfect powers: Focus on numbers that are perfect cubes or can be easily expressed as such.
• Keep terms separate: Treat the numerator and denominator individually before recombining them into a simplified fraction.

These steps help ensure the expression is as simple as possible, making further calculations straightforward and less error-prone.

Fraction Simplification

Fraction simplification involves reducing a fraction to its lowest terms. This means the numerator and the denominator have no common factors other than 1. In general, when simplifying a fraction such as \( \frac{\sqrt[3]{3}}{4} \), we ensure it cannot be reduced further.

Considerations when simplifying fractions:

• Finding common factors: Check to see if the numerator and denominator share any factors.
• Perfect cubes: Recognize numerical expressions that are perfect cubes as they simplify directly in cube root problems.

The transformation of \( \sqrt[3]{\frac{3}{64}} \) into \( \frac{\sqrt[3]{3}}{4} \) exemplifies efficient fraction simplification. Every factor must be analyzed to confirm there are no further reductions possible, ensuring clarity and accuracy in mathematical problem-solving.