Problem 39 - College Algebra with Modeling and Visualization

Question

If possible, simplify the expression by hand. If you cannot, approximate the answer to the nearest hundredth. Variables represent any real number. $$ \sqrt[5]{-7} $$

Step-by-Step Solution

Answer

The approximate value is \(-1.48\).

Step 1: Understand the Problem

The objective is to simplify the expression \( \sqrt[5]{-7} \). This expression asks for the real number that, when raised to the fifth power, gives -7.

Step 2: Consider Simplification Options

Since -7 is a negative number and we're looking for a fifth root, we can find a real number because odd roots (like the fifth root) of negative numbers are defined and give negative results.

Step 3: Determine Approximation

As there's no simple rational or integer expression that equals \( \sqrt[5]{-7} \), we will need to approximate the result to the nearest hundredth using a calculator. Calculating, we find \( \sqrt[5]{-7} \approx -1.48 \).

Step 4: Verify the Calculation

Verify the approximation by raising -1.48 to the fifth power using a calculator: \((-1.48)^5 \approx -7\). This confirms the approximation is accurate to the nearest hundredth.

Key Concepts

Simplification of Expressions

Simplification is the process of reducing an expression to its simplest form. When dealing with expressions like roots and radicals, understanding the type of root is crucial for simplification. The expression \( \sqrt[5]{-7} \) presents a challenge because 7 is not a perfect fifth power. This means we cannot reduce it further into a simple fraction or integer. However, understanding the properties of odd and even roots can guide us in trying to simplify such an expression.

• For perfect roots, the result would be an integer.
• If a number is not a perfect root, simplification might involve breaking down components, if applicable.

Here, since \(-7\) is neither a perfect power nor can be broken into components relevant to fifth powers, further simplification is not feasible. Thus, approximation is often the next step.

Approximation

When it is difficult or impossible to simplify an expression to a neat form, approximation becomes a useful tool. The aim is to find a value that, while not exact, is close enough for practical purposes. In the case of \( \sqrt[5]{-7} \), we approximate the value using a calculator.

• Calculators are essential for finding decimal approximations when manual simplifying is impractical.
• The result \( \approx -1.48 \) meets the requirement of approximating to the nearest hundredth.

Verifying this approximation involves raising \(-1.48\) to the fifth power to check if it returns a value close to \(-7\). This process ensures our approximated value is valid.

Odd Roots

Odd roots, such as the fifth root, allow us to deal with both positive and negative numbers without complications that even roots can present. The key characteristic of odd roots is that they preserve the sign of the radicand (the number under the root).

• Odd roots of negative numbers are negative.
• This differs from even roots, which do not support real results for negative radicands.

In the problem \( \sqrt[5]{-7} \), the fact that 5 is odd allows us to calculate a real, negative number that can be approximated. Without this odd root property, calculating roots of negative numbers in the real number system would be impossible.