Problem 43 - A Graphical Approach to Precalculus with Limits

Question

Answer each of the following. Explain why a positive real number must have a positive real \(n\) th root.

Step-by-Step Solution

Answer

A positive real number's n-th root is positive because a negative root would contradict the positive property of repeatedly multiplying numbers.

Step 1: Understand the Concept of n-th Root

The n-th root of a number \(a\) is a number \(b\) such that \(b^n = a\). For a positive real number, we want to determine why its n-th root must also be positive.

Step 2: Consider the Properties of Real Numbers

Recall that the product of two positive real numbers is positive. Similarly, raising a positive real number to a power (such as the n-th power) remains positive. Thus, if \(a > 0\), then \(b^n = a \) indicates \(b\) must be positive to satisfy this equality.

Step 3: Analyze the Squaring Mechanism

If \(b < 0\), then \(b^2 = b \times b\) would be positive, suggesting that repeatedly multiplying negative numbers yields a positive result if \(n\) is even, contradicting finding an n-th root when \(a > 0\). When \(n\) is odd, \(b < 0\) leads to an even more apparent contradiction since negative multiplied odd times stays negative.

Step 4: Examine the Impact of Exponential Functions

Consider real analytic functions such as exponential functions, \(b = b^{1/n}\), and note that they are defined for positive bases with any real exponent. Hence, the n-th root of any positive real number remains a positive real number by the nature of exponential expressions.

Key Concepts

Understanding Real Numbers

Real numbers encompass all the numbers that can be found on the number line, including both rational and irrational numbers. They include:

• Positive numbers, such as 1, 2, and 3.5.
• Negative numbers, like -1 and -3.5.
• Zero.

Real numbers are fundamental in mathematics because they represent quantities in everyday life, such as lengths, areas, and volumes. Positive real numbers have a straightforward property where their product is always positive. Thus, if you multiply any positive real number with another positive real number, the result is still positive. This concept is crucial when considering the n-th root, as it helps ensure that for a positive number, its n-th root must remain positive to maintain the original number's positivity.

Exploring Exponential Functions

Exponential functions involve variables in the exponent, like in the expression \( b^n \), where \( b \) is the base and \( n \) is the exponent. These functions exhibit rapid growth or decay based on the sign and value of the exponent. For example:

• When the exponent \( n \) is positive, \( b^n \) grows, assuming \( b > 1 \).
• If \( n \) is negative, then \( b^n \) shrinks, provided \( b > 1 \).

Exponential functions are crucial in defining n-th roots, as the expression \( b^{1/n} \) represents the n-th root of \( b \). If \( b \) is positive, then \( b^{1/n} \) remains defined and positive, ensuring that the operations on positive real numbers through their roots result in positive outcomes.

Properties of Roots

The properties of roots stem from their ability to undo a power operation. If \( b \) is a number such that \( b^n = a \), \( b \) is the n-th root of \( a \). Some important properties to recognize include:

• For any positive real number \( a \), its n-th root \( b \) is also positive.
• If \( n \) is an even number and \( b < 0 \), then \( b^n \) would be positive, contradicting what we seek in finding a root for a positive \( a \).
• When \( n \) is odd, \( b < 0 \) results in a negative \( b^n \), which does not equal a positive \( a \).

These properties ensure that the process of finding the n-th root for a positive real number inevitably leads to another positive real number, aligning with the product and exponential rules highlighted earlier. Hence, understanding these properties clarifies why overseeing negative roots for positive numbers doesn't fit in the realm of positive real numbers.