Problem 106 - Brief Applied Calculus

Question

For each statement, either state that it is True (and find a property in the text that shows this) or state that it is False (and give an example to show this). \(\left(x^{m}\right)^{n}=x^{m^{n}}\)

Step-by-Step Solution

Answer

False. Example: \((2^2)^3 = 64\) but \(2^{2^3} = 256\).

Step 1: Understanding the Expression

The expression given is \(\left(x^{m}\right)^{n}\). According to the laws of exponents, when you raise an exponent to another exponent, you multiply the exponents: \((x^m)^n = x^{m \cdot n}\).

Step 2: Analyzing the Statement

The statement claims that \((x^m)^n = x^{m^n}\). We need to determine if this equality holds by checking if both expressions yield the same result.

Step 3: Simplifying the Right-Hand Side

For the right-hand side, the expression is \(x^{m^n}\), which means \(m\) is raised to the power of \(n\) first, and then the base \(x\) is raised to the result. So, \(x^{m^n}\) is basically \(x\) raised to the power of \(m\) raised to the power of \(n\).

Step 4: Comparison and Example

Let's compare the two expressions using specific values for \(x\), \(m\), and \(n\). Suppose \(x = 2\), \(m = 2\), and \(n = 3\). Then, \((2^2)^3 = (4)^3 = 64\), but \(2^{2^3} = 2^8 = 256\). Since \(64 eq 256\), this demonstrates that \((x^m)^n eq x^{m^n}\).

Step 5: Conclusion

The statement is False. The property \((x^m)^n = x^{m \cdot n}\) is the correct exponent rule to apply, as demonstrated in the example where \((x^m)^n\) resulted in \(64\) but \(x^{m^n}\) resulted in \(256\). Therefore, \((x^m)^n = x^{m^n}\) does not hold true generally.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a base number, say \(x\), to an exponent, usually denoted as \(m\). This operation is expressed as \(x^m\), indicating that \(x\) is being multiplied by itself \(m\) times. Understanding exponentiation is crucial as it forms the foundation of how we simplify and manipulate expressions involving powers and scale numbers across different operations.

Some key points to remember about exponentiation include:

• The base \(x\) can be any real number.
• The exponent \(m\) can be a positive integer, a negative integer, or even a fraction.
• Exponentiation follows specific rules that govern how expressions are simplified. These rules are important when solving mathematical equations involving powers.

When you encounter an expression like \((x^m)^n\), exponentiation informs the procedure for calculating by knowing the specific rules to apply.

Mathematical Proof

A mathematical proof is a logical argument demonstrating the truth or falsehood of a mathematical statement. In the context of exponentiation and the rules governing it, a proof helps validate whether a given expression follows the standard exponent rules.

In the exercise provided, the task was to examine the truth of the statement \((x^m)^n = x^{m^n}\). Through a mathematical proof, we systematically evaluated and showed that this statement doesn't hold true by applying the correct rules.

Key aspects of constructing a mathematical proof include:

• Define what needs to be proven or disproven clearly.
• Utilize known mathematical properties or theorems that relate to the problem.
• Calculate using specific examples, such as substituting known values, to either support or refute the claim.

By proving that \((x^m)^n\) is not equal to \(x^{m^n}\), evidence is not only based on assumption but validated with examples, observing the results of applying both operations separately.

Mathematical Properties

Mathematical properties are established rules and conventions that define operations with numbers and expressions. Understanding these properties is essential for simplifying mathematical expressions accurately, especially those involving exponentiation. For instance, the original exercise dealt with the property governing the power of exponents.

Important properties related to exponentiation include:

• The Power of a Product Rule: \((xy)^m = x^m y^m\).
• The Power of a Power Rule: \((x^m)^n = x^{m \cdot n}\).
• The Power of a Quotient Rule: \((\frac{x}{y})^m = \frac{x^m}{y^m}\).

These rules ensure consistent results. For example, according to the Power of a Power Rule, applied correctly to \((x^m)^n\), it yields \(x^{m \cdot n}\), not \(x^{m^n}\). Using these properties fills the gap in our understanding when handling complex expressions, ensuring accurate outcomes.

Power of a Power Rule

The Power of a Power Rule is a specific exponentiation rule stating that when you raise one power to another, you multiply the exponents. For example, given the expression \((x^m)^n\), the rule simplifies it to \(x^{m \cdot n}\).

This rule is significant because it simplifies the process of multiplying sequences of repeated multiplications, ensuring that operations are done efficiently and correctly. By correctly applying this rule, mathematical expressions are both simplified and more manageable.

Consider applying the Power of a Power Rule:

• Identify the expression in the form \((x^m)^n\).
• Apply the rule by multiplying \(m\) and \(n\), simplifying to \(x^{m \cdot n}\).
• Verify results through substitution to confirm understanding.

In the exercise task, applying this rule showed that \((x^m)^n\) and \(x^{m^n}\) are fundamentally different by producing different results, hence confirming the falsity of the statement \((x^m)^n = x^{m^n}\). This demonstrates the practical utility and necessity of understanding the Power of a Power Rule in problem-solving.