Problem 97 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. All variables represent positive real numbers. $$ \sqrt[3]{\frac{7}{64}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{\sqrt[3]{7}}{4} \).

Step 1: Understand the problem

We need to simplify the cube root of the fraction \( \frac{7}{64} \). Variables represent positive real numbers, which ensures all operations with positive results.

Step 2: Break down the cube root

Since the expression is \( \sqrt[3]{\frac{7}{64}} \), it can be expressed as the cube root of \( \frac{7}{64} \), or \( \frac{\sqrt[3]{7}}{\sqrt[3]{64}} \).

Step 3: Simplify the denominator

The cube root of \( 64 \) is \( 4 \) because \( 4^3 = 64 \). Therefore, \( \sqrt[3]{64} = 4 \).

Step 4: Simplify the entire expression

Substituting back, \( \frac{\sqrt[3]{7}}{4} \) is the simplified form of the original expression.

Key Concepts

Understanding Cube Roots

A cube root is a number that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2 because when you multiply 2 by itself three times (2 × 2 × 2), you get 8. The symbol for cube root is \( \sqrt[3]{} \) followed by the number for which you are finding the cube root.

When dealing with fractions like \( \frac{7}{64} \), each part of the fraction can be handled separately. This means you take the cube root of the numerator \( 7 \) and the cube root of the denominator \( 64 \).

Cube roots simplify the process of solving algebraic expressions involving powers and help to break down complex numbers into more manageable parts.

The Simplification Process

Simplifying expressions involves breaking them down to their simplest form. For cube roots applied to fractions like \( \frac{7}{64} \), you can think of it as the cube root of \( 7 \) divided by the cube root of \( 64 \).

• Step 1: Identify the parts. Recognize \( 64 \) as a perfect cube because \( 4^3 = 64 \).
• Step 2: Simplify the denominator. The cube root of \( 64 \) thus becomes \( 4 \).
• Step 3: Apply the cube root to both parts. You now have \( \frac{\sqrt[3]{7}}{4} \) as the simplified expression.

Dividing or simplifying cube roots is a powerful method to make expressions easier to work with, especially when you are dealing with perfect cubes.

Working With Positive Real Numbers

Positive real numbers play a crucial role in algebra, especially when simplifying expressions like cube roots. These are numbers greater than zero and include both whole numbers and fractions.

Using positive real numbers is essential because they ensure that all operations, such as taking roots, result in valid, meaningful outcomes. Roots and powers involve working with positive numbers unless otherwise specified.

When simplifying, always consider:

• Positive numbers help retain the relationship between the parts of your expression.
• They guarantee non-negative results, which are crucial in many practical and theoretical applications.

Understanding this is key when facing algebraic equations, as it influences how you approach solutions, keeping all components in the positive realm simplifies calculations and ensures logical outcomes.