Problem 33 - A Graphical Approach to College Algebra

Question

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$-\sqrt[3]{\frac{4}{5}}$$

Step-by-Step Solution

Answer

The expression simplifies to \(-\frac{\sqrt[3]{4}}{\sqrt[3]{5}}\)."

Step 1: Identify the expression

The given expression is \(-\sqrt[3]{\frac{4}{5}}\), which represents the cube root of the fraction \(\frac{4}{5}\), multiplied by -1.

Step 2: Simplify inside the radical

In this step, observe if \(\frac{4}{5}\) can be simplified within the cube root. Since 4 and 5 have no common factors, \(\frac{4}{5}\) is already in its simplest form.

Step 3: Cube root of fraction

Apply the property of radicals which states \(\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}\). Here, \(a=4\) and \(b=5\), so we have \(-\frac{\sqrt[3]{4}}{\sqrt[3]{5}}\).

Key Concepts

Cube Roots

Cube roots are a specific type of radical notation that deal with the number which, when multiplied by itself three times, gives the original number. Just like with square roots, there are rules and properties for working with cube roots.
However, these properties can sometimes seem a bit trickier than those for square roots. For example, the cube root of 8 is 2, because when you multiply 2 by itself three times, you get 8:

• The symbol for the cube root is \(\sqrt[3]{}\).
• To find the cube root of a number, consider what number multiplied by itself three times results in the original number.
• If a number can be simplified or broken down, you can often simplify the cube root expression.

Cube roots can also apply to fractions. When dealing with fractions, you can separate the cube root of the numerator and the denominator as demonstrated in the solution:\[\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}\]
This property of cube roots simplifies the process of working with fractional radicals.

Fraction Simplification

Fraction simplification is an essential step before applying any radical operations, such as cube roots. A fraction is simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD).
This process ensures that fractions are presented in their simplest form.
For example, the fraction \(\frac{4}{5}\) in our exercise is already simplified. The numbers 4 and 5 do not share any common divisors other than 1.

• Simplify any fraction by checking for and cancelling out common factors between the numerator and the denominator.
• Look for any common factors larger than 1 to truly reduce the fraction.

By simplifying fractions first, the subsequent mathematical operations, like taking cube roots, become easier to manage and calculate.

Properties of Radicals

The properties of radicals help simplify and restructure radical expressions, making calculations more manageable. These properties apply to roots of any degree, including cube roots.
Here are some key properties:

• Product Property: \(\sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b}\)
• Quotient Property: \(\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}\)
• Power Rule: \((\sqrt[n]{a})^n = a\)

Understanding these can greatly simplify complex radicals, especially when dealing with variables or multiple radicals. In our case, we used the quotient property to manage the cube root of the fraction \(\frac{4}{5}\).
These properties allow us to break down radical expressions into simpler components, which are easier to work with during calculations.