Problem 38 - 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. $$ \sqrt[5]{-1} $$

Step-by-Step Solution

Answer

The simplification of \(\sqrt[5]{-1}\) is \(-1\).

Step 1: Understand the Expression

The expression \( \sqrt[5]{-1} \) denotes the fifth root of \(-1\). It means we are looking for a number that, when raised to the power of 5, equals \(-1\).

Step 2: Simplify the Expression

To find the fifth root of \(-1\), we note that roots of negative numbers are real if the root's degree is odd. Hence, we can simplify and determine that the fifth root of \(-1\) is \(-1\), since \((-1)^5 = -1\).

Step 3: Verify the Simplification

Verify that the simplification is correct: \((-1)^5 = (-1) \times (-1) \times (-1) \times (-1) \times (-1) = -1\). Thus, \(\sqrt[5]{-1} = -1\) is verified.

Key Concepts

Real Numbers

Real numbers encompass a broad range of numerical values. They include both rational and irrational numbers, covering everything from positive numbers and zero to negative numbers. Rational numbers are those that can be expressed as a fraction, like \(\frac{1}{2}\), while irrational numbers cannot be neatly expressed as fractions, such as \(\sqrt{2}\).
Real numbers are fundamental in various areas of mathematics and are crucial for measuring and comparing quantities.

• Positive and Negative Numbers: Real numbers include both positive numbers, like 2 or 5.7, and negative numbers, like -3 or -7.8.
• Whole and Decimal Numbers: They encompass whole numbers without fractions (0, 1, 2) and decimals which have fractional parts (3.14, -2.71).

In the context of roots and radicals, real numbers give us a base for understanding how different kinds of roots can be calculated, including fifth roots, like the one found in the given expression \(\sqrt[5]{-1}\).

Fifth Root

The concept of the fifth root involves finding a number that, when multiplied by itself four more times (totaling five times), yields the original number under the root symbol. In the expression \(\sqrt[5]{-1}\), our task is to determine the fifth root of \(-1\).

To understand whether a fifth root is possible for negative numbers, it's important to note that:

• Odd Roots of Negative Numbers: Odd roots (like the cube root or fifth root) of negative numbers are always real. This is because negative numbers raised to an odd power remain negative, thus maintaining the real property of numbers.
• Example Calculation: For \(\sqrt[5]{-1}\), the answer is \(-1\), because \((-1)^5 = -1\).

So, the fifth root of \(-1\) is indeed a real number and is equal to \(-1\). This makes fifth roots an interesting aspect of real number analysis, especially distinguishing them from even roots, which do not yield real numbers for negative radicands.

Simplifying Expressions

When simplifying expressions, especially those involving radicals and roots, the objective is to reduce the expression to its simplest, most understandable form. This often involves identifying if the expression can be simplified to a real number or another simpler radical.

In simplifying the expression \(\sqrt[5]{-1}\), we validated that:

• Simplification Check: The simplified form of \(\sqrt[5]{-1}\) is \(-1\). This outcome illustrates how recognizing the properties of roots (odd vs. even) plays a role in simplification.
• Verification: Simplifications are often verified by testing them, as shown by calculating \((-1)^5\), which indeed returns \(-1\), thus confirming the simplification's accuracy.

Simplifying expressions allows for clearer understanding and easier computation, especially when dealing with complex equations or further mathematical operations. In the given exercise, the simplification of the fifth root of \(-1\) streamlines the expression for further use or analysis.