Problem 90 - Precalculus: Functions and Graphs

Question

Replace the symbol \(\square\) with either \(=\) or \(\neq\) to make the resulting statement true, whenever the expression has meaning. Give a reason for your answer. $$a^{1 k} \square \frac{1}{a^{k}}$$

Step-by-Step Solution

Answer

\(a^k = \frac{1}{a^k}\) if \(a = 1\) or \(k = 0\); otherwise, \(a^k \neq \frac{1}{a^k}\).

Step 1: Simplify the Left Expression

The given expression is \(a^{1k}\). We know that \(a^{1k} = a^k\). So the left side of the expression simplifies to \(a^k\).

Step 2: Simplify the Right Expression

The right side of the expression is \(\frac{1}{a^k}\). This is the reciprocal of \(a^k\).

Step 3: Determine Equality or Inequality

To determine whether \(a^k\) is equal to \(\frac{1}{a^k}\), we must consider the laws of exponents. \(a^k\) and \(\frac{1}{a^k}\) are only equal if \(a^k = a^{-k}\), which implies \(a^{2k} = 1\). This is only true if \(a = 1\) or if \(k = 0\).

Step 4: Case Analysis

- If \(a = 1\), then \(a^k = 1^k = 1\) and \(\frac{1}{a^k} = \frac{1}{1} = 1\), so \(a^k = \frac{1}{a^k}\).- If \(k = 0\), then \(a^k = a^0 = 1\) and \(\frac{1}{a^k} = \frac{1}{a^0} = 1\), making both expressions equal.- For other values of \(a\) and \(k\), \(a^k eq \frac{1}{a^k}\).

Step 5: Conclusion

The statement can be true if \(a = 1\) or \(k = 0\), resulting in \(a^k = \frac{1}{a^k}\), so the correct choice is \(=\) under these conditions. Otherwise, the expression is not equal.

Key Concepts

Equality of Expressions

Equality of expressions is a foundational concept in mathematics, where two expressions represent the same value. Unlike expressions with inequalities, equal expressions can be substituted for one another. If two expressions are equal, no matter which values are substituted for the variables within them, they produce the same results.

In the exercise provided, the challenge is to determine if \(a^k\) equals \(\frac{1}{a^k}\). To find equality here, we're exploring conditions where these expressions yield the same value. Specifically, they are equal only if \(a^k\) simplifies to \(a^{-k}\), leading to \(a^{2k} = 1\).

Considering that \(a^{2k} = 1\) implies either \(a = 1\) or \(k = 0\), we see that these specific conditions ensure the expressions equate. Thus, equality of expressions is verified under these particular scenarios, showcasing how equality hinges on the values of \(a\) and \(k\). It emphasizes the concept that expressions can be conditionally equal based on the parameters involved.

Laws of Exponents

The laws of exponents are essential tools in simplifying expressions and solving equations involving exponential terms. These laws include rules like the product of powers, power of a power, and the quotient of powers, each simplifying expressions involving exponents to more manageable forms.

Relevant to our exercise, we must understand that \(a^k\) and \(\frac{1}{a^k}\) relate through the reciprocal property:

• \(a^k\) is a simple exponential expression, representing \(a\) multiplied by itself \(k\) times.
• \(\frac{1}{a^k}\) serves as the reciprocal or inverse of \(a^k\), equivalent to \(a^{-k}\) by the laws of exponents.

If \(a^k = a^{-k}\), this implies \(a^{2k} = 1\). Such condition requires specific values for \(a\) and \(k\) to satisfy the equation. This example highlights how pivotal the laws of exponents are in determining the relationship between exponential expressions, and how these laws guide us in finding equality or inequality.

Reciprocal Function

The reciprocal function is an intriguing concept in mathematics, where for any non-zero number \(a\), its reciprocal is given by \(\frac{1}{a}\). This makes it essential to understand scenarios where two expressions are inverses of each other.

In our example, \(\frac{1}{a^k}\) is the reciprocal of \(a^k\). This reciprocal transformation changes the base \(a^k\) to its inverse form, \(a^{-k}\).

• If \(a^k\) equals its reciprocal \(\frac{1}{a^k}\), then the number must satisfy the condition \(a^{2k} = 1\).
• This condition is only met when \(a = 1\) or \(k = 0\), providing instances of equality.

The reciprocal function, therefore, allows us to identify and verify scenarios where expressions might equal their inverses under specific conditions. Understanding this interchangeability and the values required for it delivers profound insights into both algebraic expressions and functional analysis.