Problem 59 - Precalculus

Question

(a) Give an example in which the result of raising a rational number to a rational power is an irrational number. (b) Give an example in which the result of raising an irrational number to a rational power is a rational number.

Step-by-Step Solution

Answer

(a) Example: \( 2^{\frac{1}{2}} = \sqrt{2} \); (b) Example: \( (\sqrt{2})^2 = 2 \).

Step 1: Understanding the Task

We need to provide examples for two scenarios: one, where a rational number raised to a rational power results in an irrational number, and two, where an irrational number raised to a rational power results in a rational number.

Step 2: Example for Part (a)

Consider the rational number 2. When raised to the rational power of \( \frac{1}{2} \), it becomes \( 2^{\frac{1}{2}} \), which is equal to \( \sqrt{2} \). The square root of 2 is well-known to be an irrational number.

Step 3: Conclusion for Part (a)

Thus, the example \( 2^{\frac{1}{2}} = \sqrt{2} \) satisfies the requirement for a rational number raised to a rational power resulting in an irrational number.

Step 4: Example for Part (b)

Take the irrational number \( \sqrt{2} \). If we raise it to the power of 2, which is a rational number, we get \( (\sqrt{2})^2 \). This simplifies to 2, which is a rational number.

Step 5: Conclusion for Part (b)

Therefore, the example \( (\sqrt{2})^2 = 2 \) shows that an irrational number raised to a rational power can result in a rational number.

Key Concepts

Rational Powers

Rational powers are essentially exponents that are expressed as fractions. If you have a number, say \( a \), and you raise it to the power of \( \frac{m}{n} \), it is equivalent to saying \( a^{\frac{m}{n}} = \sqrt[n]{a^m} \). This means that you're taking the \( n \)-th root of \( a \) and raising it to the \( m \)-th power. Raising numbers to rational powers is a way of expressing roots and combinations of roots with exponents.
Understanding how rational powers work is key to manipulating expressions and solving equations involving roots. Here are some key points:

• If \( n \) is 2, we're dealing with a square root, such as \( a^{\frac{1}{2}} = \sqrt{a} \).
• If \( m = 1\), the expression \( a^{\frac{1}{n}} \) is simply the \( n \)-th root of \( a \).

When rational powers are utilized, it often simplifies the operations on roots, making them easier to handle algebraically. Consider the example \( 2^{\frac{1}{2}} = \sqrt{2} \). Though \( 2 \) is rational, the result of this rational power is irrational, since \( \sqrt{2} \) cannot be expressed as a ratio of two integers.

Irrational Powers

Irrational powers are more complex because they involve raising a number to an exponent that cannot be expressed as a simple fraction. Unlike rational powers, they do not correspond directly to roots or simple algebraic manipulations. Typical examples include powers such as \( 2^{\sqrt{2}} \) or \( \pi^e \), which result in numbers that are usually challenging to calculate exactly.
In academic exercises, the idea of manipulating irrational powers is often to understand the nature of irrational numbers better, as well as to see surprising results, such as achieving a rational result from certain combinations:

• An example would be \( (\sqrt{2})^2 = 2 \), showing how a rational outcome can emerge from irrational bases and rational exponents.

Handling irrational powers often involves approximations and the use of specific properties of exponents, but understanding them is crucial for delving into more advanced mathematics.

Irrational Numbers

Irrational numbers are a fascinating set of numbers that cannot be expressed as the quotient of two integers. They extend the real number line beyond the rational numbers, filling in the "gaps" with numbers like \( \pi \), \( \sqrt{2} \), and \( e \).
Key characteristics of irrational numbers include the following:

• They have non-repeating, non-terminating decimal expansions.
• They can emerge from various operations, such as calculations involving square roots or certain logarithms.

Irrational numbers play a crucial role in mathematics, particularly in geometry, calculus, and number theory. For instance, the square root of any prime number is known to be irrational. When considering irrational numbers, we often employ them in mathematical proofs and theoretical applications, highlighting their unique properties and behaviors in calculations. The existence of irrational numbers is not just a mathematical curiosity; it fundamentally expands our understanding of number systems and their applications.