Problem 108 - Elementary and Intermediate Algebra

Question

Simplify each radical expression, if possible. Assume all variables are unrestricted. $$ -\sqrt[5]{-243} $$

Step-by-Step Solution

Answer

The simplified expression is \(-3\).

Step 1: Simplify the Radicand

Begin by focusing on the expression under the radical: \(-243\). Recognize that \(-243\) can be rewritten as \(-1 \times 243\), which helps us separate the negative sign for simplification.

Step 2: Identify Fifth Roots

Since we have a fifth root, aim to simplify: \(\sqrt[5]{243}\). Notice that \(243\) is actually equal to \(3^5\). Thus, \(-243 = -1 \times 3^5\).

Step 3: Apply the Radical Property

Use the property \(\sqrt[n]{a \times b} = \sqrt[n]{a} \times \sqrt[n]{b}\) to split the radical: \(\sqrt[5]{-1 \times 3^5} = \sqrt[5]{-1} \times \sqrt[5]{3^5}\).

Step 4: Calculate Individual Fifth Roots

Solve each part: \(\sqrt[5]{-1}\) is \(-1\) because \((-1)^5 = -1\), and \(\sqrt[5]{3^5}\) is \(3\) because \(3^5 = 243\).

Step 5: Multiply Results

Now multiply the two results: \(-1 \times 3 = -3\). Therefore, \(-\sqrt[5]{-243} = -3\).

Key Concepts

Understanding Fifth Roots

When you encounter a radical with the index of 5, you are dealing with a fifth root. Understanding fifth roots is crucial, especially when simplifying radical expressions. A fifth root of a number is a value that, when multiplied by itself five times, equals the original number.

For example, the fifth root of 243 is 3 because multiplying five 3s together (i.e., \(3 \times 3 \times 3 \times 3 \times 3 = 243\)) results in 243. Similarly, the fifth root of -1 is -1, since \((-1) \times (-1) \times (-1) \times (-1) \times (-1) = -1\).

Key points about fifth roots:

• They work similarly to square and cube roots but require five factors.
• Negative numbers can have fifth roots because multiplying an odd number of negative factors results in a negative number.
• To solve expressions with fifth roots, identify numbers that are powers of 5.

Simplifying Radicals Made Simple

Simplifying a radical involves breaking down the number inside the radical sign into its prime factors, particularly focusing on those that are perfect powers matching the index of the radical.

In the example of simplifying \(\sqrt[5]{-243}\), we look for a number that forms a perfect power of 5. Notice that 243 can be written as \(3^5\). By expressing 243 as a power of 3, we can clearly see how to simplify the expression.

Steps to simplify radicals:

• Break down the radicand (the number inside the radical sign) into its prime factors.


• Identify perfect powers that match the radical index number, so they can be simplified out.


• Apply any necessary radical properties to help in simplification.

Simplifying radicals allows you to express complex radicals into more straightforward terms, often making calculations or further algebraic manipulations much easier.

Useful Properties of Radicals

Understanding the properties of radicals is essential for simplifying radical expressions effectively. These properties provide rules that let you manipulate and simplify radicals easily.

Let's look at a few key properties:

• **Product Property of Radicals**: \(\sqrt[n]{a \times b} = \sqrt[n]{a} \times \sqrt[n]{b}\). This property helps split a radical into separate radicals, making it easier to handle each part independently.


• **Power Property of Radicals**: \(\sqrt[n]{a^n} = a\) if \(a\) is a real number and \(n\) is an odd integer. This occurs because raising to a power and taking a root are inverse operations.


• **Negative Under the Radical**: When dealing with odd roots, negative numbers are allowable, such as \(\sqrt[5]{-1} = -1\), because multiplying an odd number of negative numbers results in a negative outcome.

By mastering these properties, you gain powerful tools to break down complex radical expressions into simpler, more manageable terms, just like in our given example. The properties streamline the process of extracting real number solutions from otherwise complicated radical equations.