Problem 20 - Algebra A Combined Function

Question

Use the quotient rule to simplify. Assume that all variables represent positive real numbers. $$ \sqrt[3]{\frac{3}{64}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{\sqrt[3]{3}}{4} \).

Step 1: Understand the Exercise

The exercise asks you to simplify the expression \( \sqrt[3]{\frac{3}{64}} \). This is a cube root of a fraction.

Step 2: Apply Cube Root to the Numerator and Denominator

The cube root of a fraction can be expressed as the cube root of the numerator divided by the cube root of the denominator:\[\sqrt[3]{\frac{3}{64}} = \frac{\sqrt[3]{3}}{\sqrt[3]{64}}\]

Step 3: Simplify the Cube Root of the Denominator

The cube root of 64 is 4, since \(4 \times 4 \times 4 = 64\). Therefore, you have:\[\frac{\sqrt[3]{3}}{4}\]

Step 4: Express the Final Answer

Since \( \sqrt[3]{3} \) cannot be further simplified, the simplified form of the expression is:\[\frac{\sqrt[3]{3}}{4}\]

Key Concepts

Cube Root

The cube root is an operation that asks you to find a number which, when multiplied by itself three times, gives the original number. For example, the cube root of 64 is 4 because multiplying 4 by itself twice,

• 4 × 4 × 4 = 64

When dealing with cube roots, you're essentially undoing the cube operation. Cube roots are denoted as \( \sqrt[3]{x} \), where \( x \) is the number you need to "un-cube." In this specific exercise, we are tasked with taking the cube root of a fraction, \( \sqrt[3]{\frac{3}{64}} \). Understanding cube roots is crucial for simplifying expressions like these in math. It helps you break down expressions into simpler parts that are easier to manage.

Simplifying Expressions

When simplifying expressions, your goal is to reduce them to their simplest form without changing their value. Simplifying can make both calculations and comprehension easier. For the expression \( \sqrt[3]{\frac{3}{64}} \), simplification involves breaking down the expression into more manageable parts. We do this by applying the cube root individually to the numerator and the denominator, as shown:

• \( \sqrt[3]{\frac{3}{64}} = \frac{\sqrt[3]{3}}{\sqrt[3]{64}} \)

In mathematical expressions, simplifying is widely used to help solve equations, to understand complex functions, and to perform quick, accurate calculations. Each simplification step should maintain the original essence and properties of the expression.

Numerator and Denominator

The terms numerator and denominator refer to the two components of a fraction. The numerator is the number above the fraction line, while the denominator is the number below it. In the fraction \( \frac{3}{64} \), 3 is the numerator, and 64 is the denominator.

• The numerator indicates how many parts of the whole are being considered.
• The denominator explains into how many parts the whole is divided.

Understanding these parts is crucial when simplifying fractions because operations like finding roots need to be applied to both. This brings clarity when performing operations like division or multiplication in algebraic expressions. In the expression \( \sqrt[3]{\frac{3}{64}} \), by computing these parts separately, you can simplify the expression step-by-step.

Positive Real Numbers

Positive real numbers are the set of all real numbers greater than zero. They are used ubiquitously in math, ensuring that certain mathematical operations result in real-number outcomes.When working with problems that require assumptions of positivity, as mentioned in your exercise, it allows for all operations performed to stay within the real number system. It avoids complications that can arise from dealing with negative values, such as obtaining non-real results from certain roots or logarithms.In this exercise, assuming positive real numbers for \( \sqrt[3]{\frac{3}{64}} \) guarantees our calculations have meaningful, real results. We can confidently apply the cube root and quotient rule without worrying about undefined or complex outcomes. This concept simplifies the process and ensures clarity in results.