Problem 89 - Intermediate Algebra

Question

Evaluate each radical expression, if possible, without using a calculator. See Example 8. $$ -\sqrt[5]{243} $$

Step-by-Step Solution

Answer

The expression evaluates to \\-3\\.

Step 1: Identify the Expression

We need to evaluate the expression \(-\sqrt[5]{243} \), which is asking for the negative fifth root of 243.

Step 2: Understand Root and Power

The fifth root of a number means finding a number that, when raised to the power of 5, gives the original number. We need to find \( x \) such that \( x^5 = 243 \).

Step 3: Find Base Number

To determine this, we start by checking small integers to see which one raised to the fifth power equals 243. With practice, known results, or calculation: \( 3^5 = 243 \) is correct, because \( 3*3*3*3*3 = 243 \).

Step 4: Consider the Negative Sign

The original expression has a negative sign in front. Thus, the expression \(-\sqrt[5]{243} \) becomes \(-3\) because the expression asks for the negative fifth root.

Step 5: Verify

Verify the result by ensuring that raising \(3\) to the fifth power gives 243, which confirms that the fifth root of 243 is indeed 3 and the negative fifth root is \(-3\).

Key Concepts

Fifth Roots

When we talk about the fifth root of a number, we're essentially looking for a number that you can multiply by itself five times to achieve the original number. To simplify, if you have a number \( x \) and you raise it to the fifth power, written as \( x^5 \), you should get the original number we are working with. If we say the fifth root of 243, we mean: what number multiplied by itself five times equals 243?

Finding such roots can be done by testing small integers. In our example with 243, we found that \( 3^5 = 243 \). This verification is crucial to confirm that 3 is indeed the fifth root. Once we know the fifth root of a number, it's simple to find the root of its negative counterpart by just directing the result with the required sign that was initially part of the expression, as in -3 for -\( \sqrt[5]{243} \).

• Fifth root reverses the action of raising a number to the power of 5.
• Verify by checking small successive integers.
• Use known powers and calculations as shortcuts.

Integer Powers

Understanding integer powers is key in simplifying radical expressions. Integer powers like \( x^5 \), \( x^2 \), or any \( x^n \), represent multiplying \( x \) by itself \( n \) times. For instance, \( 3^5 \) means 3 multiplied by itself 5 times – \(3 \times 3 \times 3 \times 3 \times 3\).

Integer powers reflect growing a number exponentially and understanding this helps make finding roots intuitive. In recognizing powers, familiarity with common powers assists greatly. A firm grasp on powers of numbers such as 2, 3, or 5 can simplify many calculations and checks.

• Integer powers are repeated multiplications.
• They indicate exponential growth of numbers.
• Common powers can be memorized to simplify roots.

Simplifying Radicals

Simplifying radicals involves reducing a root to its simplest form. Let's consider a radical like \( \sqrt[5]{243} \). When you solve for the fifth root as we did above, you simplify the expression from a complex looking radical to just a plain integer—3 in this case.

This process involves identifying the appropriate base number that satisfies the radical's requirement. Simplifying not only gives a clearer expression but also aids in easier computation for further mathematical operations.

• Expression simplification leads to concise results.
• Ensures expressions are in their simplest, most comprehensible form.
• Reduces computational complexity in handling radicals.