Problem 64 - College Algebra Essentials

Question

Evaluate each expression in Exercises \(55-66,\) or indicate that the root is not a real number. $$\sqrt[5]{(-2)^{5}}$$

Step-by-Step Solution

Answer

\(\sqrt[5]{(-2)^{5}} = -2\)

Step 1: Calculate the base

First, calculate the base expression, \((-2)^5\), which is -32.

Step 2: Evaluate the Root

Now, calculate the fifth root of -32. Since the index (the root number) is odd and the base is negative, the result is a real number.

Step 3: Solution

Therefore, \(\sqrt[5]{(-2)^{5}} = -2\).

Key Concepts

Fifth Roots

The fifth root of a number refers to a value that, when multiplied by itself five times, equals the original number. Consider the expression \( \sqrt[5]{x} \). Here, you are finding a number that, when raised to the power of five, results in \( x \).

If x is a positive number, its fifth root will also be positive. If x is negative, you'll need to see if the root is possible and real by considering its properties, like the index of the root.

• The concept of roots is an extension of inverse operations. Just as exponentiation raises a number to a power, the fifth root reverses this process.
• Fifth roots can be fractional or whole numbers, depending on the value inside the root.

To put this into practice with our exercise, \( \sqrt[5]{(-32)} = -2 \) because \( (-2)^5 = -32 \), confirming that -2 is indeed the fifth root of -32.

Odd Indices

Odd indices refer to the power or the root number, like 3 or 5, in expressions like \( x^3 \) or \( \sqrt[5]{x} \). Odd indices play an important role in identifying whether real number solutions exist with negative bases.

One of the unique attributes of odd indices is that they accommodate both positive and negative numbers under the root:

• For odd powers, raising a negative number still results in a negative product. For example, \( (-2)^3 = -8 \).
• This concept works with roots, too. A fifth root with an odd index will yield a real number even if the base is negative.

Odd indices ensure that real solutions exist, making them distinct from even indices, which generally don't lead to real solutions for negative bases.

Negative Bases

Negative bases in mathematical expressions are numbers less than zero used as the base for powers. In expressions like \((-2)^n\), the sign and real number status depend on \( n \), or the index. With our example, \((-2)^5\), it's useful to see how negative bases behave:

• If the index is even: Practically, \((-2)^4 = 16\), because multiplying four negative numbers results in a positive number.
• If the index is odd, it remains negative: As seen with \((-2)^5 = -32\), odd numbers retain the negative sign.

Working with negative bases requires careful attention to the indices since the results will vary significantly. Identify the index to determine how the base sign affects the final product.

Exponentiation

Exponentiation is a mathematical operation involving numbers called the base and the exponent (or power). It is denoted in the form \( a^n \), where \( a \) is the base and \( n \) is the exponent. The operation calculates multiplying the base \( a \), \( n \) times.

In our problem

• This step involves \((-2)^5\), which instructs to multiply -2 by itself five times.
• Completing this calculation shows the product as -32.

This operation is fundamental to understanding deeper algebraic operations such as roots, which reverse or undo exponentiation to find initial values from their multiples.