Problem 30 - Intermediate Algebra

Question

Find each root. Assume that all variables represent nonnegative real numbers. $$ \sqrt[5]{-243} $$

Step-by-Step Solution

Answer

The fifth root of \(-243\) is \(-3\).

Step 1: Identify the Expression

We need to find the fifth root of \(-243\). The expression is \(\sqrt[5]{-243}\). Since we're dealing with real numbers, we must consider whether there is a real fifth root for a negative number.

Step 2: Evaluate the Sign of the Root

For odd roots, such as cube roots or fifth roots, we can take the root of a negative number and get a negative result. This is unlike even roots, where a negative number under the root would be undefined in the reals.

Step 3: Simplify the Expression

Since \(-243\) is a negative number, the fifth root of \(-243\) can be found by recognizing the pattern of powers. We know that \(243 = 3^5\), therefore \(-243 = (-3)^5\). Thus, \(\sqrt[5]{-243} = -3\).

Key Concepts

Understanding the Fifth Root

When we say "fifth root," we are referring to the number that, when multiplied by itself five times, gives the original number. In other words, the fifth root of a number \(a\) is a number \(b\) such that \(b^5 = a\). For example, if we want to find the fifth root of 32, we are essentially looking for a number that when raised to the power of five equals 32. In this case, the answer is 2 because \(2^5 = 32\). Finding a fifth root is very similar to finding square roots or cube roots, but it extends to five numbers multiplied together. When dealing with negative numbers, as we will see, the rules can be interesting.
Always remember that the concept of roots helps us solve equations and understand relationships between numbers. Determining the fifth root involves reversing the process of exponentiation, making it a powerful tool in mathematical calculations.

Dealing with Negative Numbers in Roots

Negative numbers can be tricky when dealing with roots. Here's a simple rule of thumb:

• For even roots (like square roots, four roots, etc.), a negative number under the root does not have a real root.
• For odd roots (like cubic roots, fifth roots, etc.), negative numbers do indeed have a real root.

This is because an even power or root always results in a positive number, meaning negative numbers cannot be reached without involving imaginary numbers. However, with odd roots, a negative number under the root is valid, and the result will also be negative. Take the problem \(\sqrt[5]{-243}\), for instance. Since 5 is an odd number, the fifth root of -243 exists and is a real number. We can simplify \(-243\) into \((-3)^5\). Consequently, \(\sqrt[5]{-243} = -3\), a perfect example of how negative numbers behave under odd roots.
Recognizing when and how negative numbers can be simplified with roots is crucial for navigating mathematical problems effectively.

Understanding Odd Roots and Their Properties

Odd roots have special properties that distinguish them from even roots. With odd roots like cube roots or fifth roots, you can find a real root for any real number, positive or negative. Here’s why:

• Odd roots can transform a negative number into a negative root and a positive number into a positive root.
• This property means the mathematical world of real numbers remains accessible, compared to the challenges posed by even roots.

Let's apply this to our solution. To find \(\sqrt[5]{-243}\), we employed the property of odd roots that allows us to simplify a negative number by recognizing its power structure. Since \(-243 = (-3)^5\), the task reduces to identifying the number \(-3\) whose fifth power returns the original \(-243\). Hence, understanding odd roots enables us to solve problems with negative numbers more intuitively, emphasizing the broad possibilities allowed by odd exponents in root calculations.