Problem 60 - Precalculus

Question

Give an example of irrational numbers a and b such that the indicated expression is (a) rational and (b) irrational. Can an irrational number raised to an irrational power yicld an answer that is rational? This problem shows that the answer is "yes." (However, if you study the following solution very carefully, you'll see that even though we've answered the question in the affirmative, we've not pinpointed the specific case in which an irrational number raised to an irrational power is rational.) (a) Let \(A=(\sqrt{2})^{\sqrt{2}} .\) Now, either \(A\) is rational or \(A\) is irrational. If \(A\) is rational, we are done. Why? (b) If \(A\) is irrational, we are done. Why? Hint: Consider \(A^{\sqrt{2}}\)

Step-by-Step Solution

Answer

Yes, \((\sqrt{2})^{\sqrt{2}}\) is either rational itself or results in a rational number when squared.

Step 1: Define the Problem

We need to find two irrational numbers, \( a \) and \( b \), such that \( a^b \) is either rational or irrational as indicated.

Step 2: Consider \( (\sqrt{2})^{\sqrt{2}} \)

First, take \( A = (\sqrt{2})^{\sqrt{2}} \). Check the nature of \( A \): it could be rational or irrational.

Step 3: Case 1 - If \( A \) is Rational

If \( A \) is rational, then we have found our example without any further steps. Here \( a = \sqrt{2} \) and \( b = \sqrt{2} \) make the expression rational.

Step 4: Case 2 - If \( A \) is Irrational

If \( A \) is irrational, we then need to raise \( A \) again by \( \sqrt{2} \): \( A^{\sqrt{2}} = ((\sqrt{2})^{\sqrt{2}})^{\sqrt{2}} = (\sqrt{2})^{2} \).

Step 5: Calculate \( (\sqrt{2})^{2} \)

Calculate \((\sqrt{2})^{2} = 2\). Since 2 is rational, if \( A \) is irrational, then an irrational number raised to an irrational power results in a rational number.

Step 6: Conclusion

We have shown that either \( (\sqrt{2})^{\sqrt{2}} \) itself is rational, or it is irrational but when further raised to the \( \sqrt{2} \), it becomes rational as \((\sqrt{2})^2 = 2\).

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction of two integers, where the numerator is an integer and the denominator is a non-zero integer. This means they include simple numbers like 1/2, 3, and -4/7. Any number that can be written in this fractional form is considered rational.

Key characteristics of rational numbers include:

• They can be either positive or negative.
• They encompass whole numbers and integers since integers can also be considered fractions (e.g., 3 can be written as 3/1).
• When expressed in decimal form, rational numbers either terminate or repeat.

Understanding rational numbers is important because they form the basis of many mathematical principles, making them foundational in mathematics.

Exponents

Exponents are a way to represent repeated multiplication of a number by itself. In the expression such as \( a^b \), "a" is the base and "b" is the exponent, which tells us how many times to multiply the base by itself.

For example, in \( 2^3 \), the base is 2 and the exponent is 3, which means \( 2 \times 2 \times 2 = 8 \). Exponents follow certain rules:

• Multiplying powers with the same base: \( a^m \times a^n = a^{m+n} \).
• Dividing powers with the same base: \( \frac{a^m}{a^n} = a^{m-n} \).
• Raising a power to a power: \( (a^m)^n = a^{m \times n} \).

Understanding exponents is crucial because they simplify complex multiplication problems and help in expressing very large or very small numbers in a compact form.

Irrational Exponents

Irrational exponents occur when the exponent is an irrational number, meaning it cannot be expressed as a simple fraction. Expressions with irrational exponents can lead to surprising results, such as yielding rational numbers.

A classic example is considering \( (\sqrt{2})^{\sqrt{2}} \). This expression can be rational or irrational, which showcases the unique nature of irrational exponents.

While irrational exponents might seem intimidating, they follow the same properties and rules as rational exponents. They introduce more complex calculations, but with practice, they can be managed effectively.

• For any positive number \( a \), raising it to an irrational exponent involves understanding the concept of roots and powers simultaneously.
• An expression like \( a^{\sqrt{b}} \) tends to have non-repeating and non-terminating decimal expansions if not directly resolvable.

Mastering irrational exponents enhances problem-solving skills and enriches the comprehension of exponential expressions.

Mathematical Reasoning

Mathematical reasoning involves using logical thinking to solve problems and arrive at conclusions based on available information and principles. In mathematics, reasoning is fundamental for proving statements, such as whether an expression is rational or irrational.

In the given problem, reasoning allows us to deduce the behavior of expressions with irrational numbers and exponents:

• Start by assuming different possibilities, such as "either \( A \) is rational or \( A \) is irrational." This helps in structuring the reasoning process.
• Analyze each case methodically, as done in the example: if \( A = (\sqrt{2})^{\sqrt{2}} \) and is assumed rational, then no further calculation is needed. If assumed irrational, examining \( A^{\sqrt{2}} \) helps elucidate the outcome.
• Seek patterns and relationships, which are integral to establishing valid arguments in mathematics.

Strong mathematical reasoning skills are vital for tackling complex problems, ensuring the learner can develop robust solutions.