Problem 18 - Algebra for College Students

Question

Evaluate each of the following. For example, \(\sqrt{25}=5\). \(\sqrt[3]{-\frac{8}{27}}\)

Step-by-Step Solution

Answer

The cube root of \(-\frac{8}{27}\) is \(-\frac{2}{3}\).

Step 1: Understand the Expression

The given expression is \( \sqrt[3]{-\frac{8}{27}} \). We need to find the cube root of the fraction \(-\frac{8}{27}\). The cube root operation is asking what number, when multiplied by itself three times, will give \(-\frac{8}{27}\).

Step 2: Determine the Cube Root of the Numerator

The numerator is \(-8\). Let's find which number cubed gives \(-8\). The number is \(-2\) because \((-2) \times (-2) \times (-2) = -8\). Therefore, the cube root of \(-8\) is \(-2\).

Step 3: Determine the Cube Root of the Denominator

The denominator is \(27\). Let's find which number cubed gives \(27\). The number is \(3\) because \(3 \times 3 \times 3 = 27\). Therefore, the cube root of \(27\) is \(3\).

Step 4: Combine the Cube Roots

Now that we have the cube roots of the numerator and denominator, we can combine them. The expression can be rewritten as \(\frac{\sqrt[3]{-8}}{\sqrt[3]{27}} = \frac{-2}{3}\). Thus, the cube root of \(-\frac{8}{27}\) is \(-\frac{2}{3}\).

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers. They take the form \( \frac{a}{b} \) where both \(a\) (the numerator) and \(b\) (the denominator) are integers, and \(b\) is not zero. Rational numbers include fractions, integers, and mixed numbers. They are called rational because they involve ratios. Understanding this concept is crucial because it helps us handle different operations involving rational numbers, such as finding cube roots or solving equations.

• Examples of rational numbers include \( \frac{3}{4}, -\frac{1}{2}, \) and \(5\) (which can be written as \( \frac{5}{1} \)).
• If a number cannot be expressed as a fraction (like \( \sqrt{2} \)), it is not rational.
• To simplify working with rational numbers, make sure the fractions are in their simplest form.

Since the operation involves both negative and positive rational numbers, always remember the sign rules: two like signs make a positive, and two unlike signs make a negative. This will help in solving operations correctly.

Numerator and Denominator

In any fraction, the numerator and denominator are key components. The numerator is the number above the fraction line, and it indicates how many parts of the whole are being considered. The denominator, on the other hand, is the bottom part and shows how many equal parts the whole is divided into. In the fraction \( \frac{8}{27} \):

• The numerator is 8, meaning there are 8 parts being considered.
• The denominator is 27, signifying the whole has been divided into 27 equal parts.

It's also essential to consider the signs of these components. If the numerator or denominator is negative, it changes the sign of the entire fraction. For example, \( -\frac{8}{27} \) represents a negative value due to the negative sign on the numerator. When performing operations like cube roots on such a negative fraction, it's important to handle the signs correctly.

Fraction Operations

Performing operations on fractions, such as finding their cube roots, involves a clear understanding of what each operation entails. Taking a cube root is the opposite of raising a number to the third power. For any fraction \( \frac{-8}{27} \), the process involves:

• Finding the cube root of the numerator: Determine a number which when cubed results in \(-8\). In this case, \(-2\) because \((-2) \times (-2) \times (-2) = -8\).
• Finding the cube root of the denominator: Find what number cubed gives 27. Here, \(3\) because \(3 \times 3 \times 3 = 27\).
• Combining these cube roots to form a new fraction. Here, it’s \(\frac{-2}{3}\).

Sometimes, it's necessary to simplify fractions after finding the cube roots, although in this particular case, \(\frac{-2}{3}\) is already in its simplest form. Clearly understanding these operations allows one to solve various math problems involving fractions efficiently.