Problem 76 - Intermediate Algebra

Question

Simplify each cube root. See Example 6. $$ \sqrt[3]{\frac{125}{64}} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{5}{4} \).

Step 1: Identify the Cube Roots Separately

The cube root can be separated into the cube root of the numerator and the cube root of the denominator. Thus, \( \sqrt[3]{\frac{125}{64}} = \frac{\sqrt[3]{125}}{\sqrt[3]{64}} \).

Step 2: Simplify the Cube Root of the Numerator

Identify the prime factors of 125. Since 125 is \(5^3\), the cube root of 125 is 5. Thus, \( \sqrt[3]{125} = 5 \).

Step 3: Simplify the Cube Root of the Denominator

Identify the prime factors of 64. Since 64 is \(4^3\) or \(2^6\), the cube root of 64 is 4. Thus, \( \sqrt[3]{64} = 4 \).

Step 4: Combine the Simplified Parts

Place the simplified cube roots from the numerator and denominator back together. Hence, \( \frac{\sqrt[3]{125}}{\sqrt[3]{64}} = \frac{5}{4} \). This is the simplified form of the original expression.

Key Concepts

Understanding Fraction Simplification

Fraction simplification is an essential mathematical process that involves reducing a fraction to its simplest form. This is done by dividing the numerator and the denominator by their greatest common factor (GCF). Simplifying fractions helps to make calculations more manageable and expressions more understandable. It’s like cleaning up an equation to reveal its simplest essence.

• **Step 1**: Find the GCF of both the numerator and the denominator.
• **Step 2**: Divide both the numerator and the denominator by the GCF.
• **Result**: The fraction is now in its simplest form, which is easier to work with.

By learning to simplify fractions effectively, you can handle mathematical problems with greater ease. Whether dealing with basic arithmetic or more complex algebraic equations, simplifying fractions is a vital skill.

Prime Factorization Basics

Understanding prime factorization is key to breaking down numbers into their building blocks. It involves expressing a number as a product of its prime factors. This is particularly useful in simplifying cube roots and other radical expressions.

• **Prime Factorization**: Start by dividing the number by the smallest prime (2) and proceed to higher primes as necessary.
• **Importance**: Helps in identifying perfect cubes or squares, as well as simplifying roots and fractions.

For example, in the cube root simplification of 125 and 64, we recognize 125 as being composed of three 5’s (i.e., \(125 = 5^3\)). Similarly, 64 is composed of six 2’s, or \(64 = 2^6\). These factorizations allow us to simplify the cube roots easily.

Numerator and Denominator in Fractions

The numerator and denominator are the two fundamental parts of a fraction. They play critical roles in operations involving fractions. Understanding the distinction and relationship between the two helps in comprehending fraction operations more deeply.

• **Numerator**: The top part of a fraction representing how many parts of the whole or group are being considered.
• **Denominator**: The bottom part of a fraction representing the total number of equal parts that make up a whole.

In the simplification of \(\sqrt[3]{\frac{125}{64}}\), breaking the cube root into separate operations on the numerator \(\sqrt[3]{125}\) and the denominator \(\sqrt[3]{64}\), helped simplify the expression to \(\frac{5}{4}\). This illustrates how dealing with these parts individually can simplify complex expressions.

Introduction to Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the backbone of algebra, allowing for generalizations and problem-solving in mathematics. Expressions can often require simplification to become more useful.

• **Components**: Can contain numbers, variables (like \(x\) or \(y\)), and operation signs like plus or minus.
• **Simplification**: Involves reducing expressions by combining like terms or factorizing, similar to simplifying fractions or radicals.

In the exercise provided, although the numbers are concrete, the principle remains—you apply rules of simplification to promote clearer understanding and more manageable calculations. Recognizing patterns such as perfect cubes supports this process in algebraic contexts.