Problem 40 - Algebra A Combined Function

Question

Find each root. Assume that all variables represent nonnegative real numbers. $$ \sqrt[5]{-1} $$

Step-by-Step Solution

Answer

The 5th root of -1 is -1.

Step 1: Understanding the Problem

We need to find the 5th root of -1. This means we are looking for a number which, when multiplied by itself 5 times, results in -1.

Step 2: Analyzing the Root and the Negative Number

The problem asks for the 5th root of -1. The number -1 can be expressed as negative one raised to the power of one, -1 = -1^1. A negative number to an odd root can also be negative. Therefore, an odd root like a 5th root will still result in a negative number.

Step 3: Solving the Calculation Directly

We calculate the 5th root of -1 by realizing that it can be simplified directly: (-1)^{1/5} is the solution. Since -1 raised to an odd power is still -1, and the 5th root is an odd root, the result is simply -1.

Key Concepts

Negative Numbers

Negative numbers often bring confusion, but they follow particular rules that make them easy to work with when understood. A negative number is any number less than zero, marked with a minus sign (-). These numbers are essential in mathematics because they allow us to describe values that indicate loss, deficiency, or opposites.
For instance, if the temperature drops by 5 degrees from zero, we describe that as -5 degrees.

• In multiplication and division, a negative number times a positive number yields a negative result: \( -5 \times 3 = -15 \).
• If two negative numbers are multiplied or divided together, the result is positive: \( -4 \times -2 = 8 \).
• Negative numbers also behave interestingly with powers, which is crucial when dealing with roots.

Odd Roots

Odd roots, like the 5th root or cube root, have unique properties that distinguish them from even roots.
An odd root of a number is another number which, when raised to the odd-numbered power, results in the original number. For example, the 5th root of a number \( x \) is a number \( y \) such that \( y^5 = x \).

• Odd roots can accept negative numbers because raising a negative number to an odd power keeps it negative.
  Hence, the odd root of a negative number is negative.
• This means the 5th root of \(-1\) is simply \(-1\), because \((-1)^5 = -1\).
• Odd roots behave continuously over the real number line, encompassing positive, zero, and negative values.

Real Numbers

Real numbers are the set of numbers that include all the rational and irrational numbers. They can be represented on a number line, encompassing every number you typically use in daily life - from fractions and decimals to negatives and zero.

• Real numbers include:
  • Integers (e.g., -3, 0, 1)
  • Fractions and decimals (e.g., 1/2, 3.14)
  • Irrational numbers (e.g., \( \sqrt{2} \), \( \pi \))
• In the context of roots, real numbers assure us that operations like square roots or cube roots have real solutions, as long as we adjust for even roots of negative numbers (which require imaginary numbers).
• The concept of real numbers helps us to distinguish which roots result in real solutions and which are not applicable without extending into complex numbers.