Problem 32 - College Algebra with Corequisite Support

Question

For the following exercises, simplify each expression. \(\sqrt[5]{\frac{-32}{243}}\)

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{2}{3}\).

Step 1: Identify the Expression

The given expression is \(\sqrt[5]{\frac{-32}{243}}\). This is a fifth root of a fraction. Our goal is to simplify this expression.

Step 2: Simplify the Numerator

The numerator is \(-32\), which is equal to \(-2^5\). Since we are taking the fifth root of \(-2^5\), the fifth root of \(-32\) is \(-2\).

Step 3: Simplify the Denominator

The denominator is \(243\), which is equal to \(3^5\). Therefore, the fifth root of \(243\) is \(3\).

Step 4: Combine the Results

Combine the simplified fifth roots of the numerator and denominator to obtain the simplified expression: \(-\frac{2}{3}\).

Key Concepts

Fifth Root

A fifth root is a type of radical expression where you need to find a number that, when multiplied by itself five times, gives the original number. For example, in the expression \( \sqrt[5]{x} \), we are looking for a number \( y \) such that \( y^5 = x \). To better understand this, let's consider our original expression \( \sqrt[5]{\frac{-32}{243}} \). Here, we are searching for numbers that satisfy

• For the numerator: a number that gives \(-32\) when raised to the fifth power.
• For the denominator: a number that gives \(243\) when raised to the fifth power.

Once we find these numbers, we'll have their respective fifth roots, and consequently, we can simplify the expression. The concept of the fifth root allows us to break down seemingly complex expressions into manageable pieces.

Numerator and Denominator Simplification

In fraction simplification, understanding the individual parts—the numerator and the denominator—is crucial. Typically, you simplify each part separately and then combine the results to simplify the entire fraction.Let's work through our original expression, \( \sqrt[5]{\frac{-32}{243}} \):

• Numerator (-32): We notice that \(-32\) can be expressed as \(-2^5\). Using our knowledge of the fifth root, the fifth root of \(-2^5\) is \(-2\) because \((-2)^5 \) results in \(-32\).
• Denominator (243): Similarly, \(243\) can be broken down to \(3^5\). So, finding the fifth root of \(243\) gives us \(3\).

Understanding how to express the numerator and denominator as powers greatly aids in retrieving their fifth roots quickly, thereby simplifying the problem at hand.

Fraction Simplification

Simplifying fractions involves dividing the numerator and the denominator by their greatest common factor, though in this context, we rely on understanding radical operations instead. The rule of thumb is to simplify each part of the fraction and then reassemble it.Reconsidering our expression \( \sqrt[5]{\frac{-32}{243}} \), we simplified the numerator to \(-2\) and the denominator to \(3\):

• The simplified fifth root of the numerator \(-32\) is \(-2\).
• The simplified fifth root of the denominator \(243\) is \(3\).

By combining these results, you conclude with the simplified fraction \(-\frac{2}{3}\). This approach makes the process less daunting. You isolate each component, apply the appropriate math operations, and assemble your conclusions into a simplified expression. Through practice, these steps will become more intuitive.