Problem 90 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 85-90: 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

\(\square = \neq\) since \(a^{1/k} \neq \frac{1}{a^k}\) unless \(a = 1\).

Step 1: Analyze both expressions

The expression on the left, \(a^{1/k}\), represents the \(k^{th}\) root of \(a\), which can also be written as \(\sqrt[k]{a}\). The expression on the right, \(\frac{1}{a^k}\), represents the reciprocal of \(a\) raised to the power \(k\). We need to compare these two expressions based on the properties of exponents and roots.

Step 2: Evaluate the conditions for equality

For the two expressions to be equal, \(a^{1/k} = \frac{1}{a^k}\), it must be true that \(a^{1/k} \times a^k = 1\). Simplifying this, we get \(a^{1/k + k} = a^{k/k + k^2/k} = a^{(1 + k^2)/k}\). This can only be equal to 1 when \(a = 1\) or when the exponent is zero, \((1 + k^2)/k = 0\).

Key Concepts

Powers and Roots

Understanding powers and roots is crucial for mastering exponents and radicals. A power refers to the number of times a number is multiplied by itself. For example, in the expression \(a^k\), \(a\) is the base, and \(k\) is the exponent or power, indicating how many times \(a\) is used as a factor.
Roots, on the other hand, are the inverse operation of exponentiation. The \(k^{th}\) root of a number \(a\) is a value that, when raised to the \(k\), gives \(a\). This is often represented as \(a^{1/k}\) or \(\sqrt[k]{a}\).
In the exercise, \(a^{1/k}\) is the \(k^{th}\) root of \(a\), providing a way to "undo" the operation of raising \(a\) to the power \(k\).
By grasping the interchange between powers and their corresponding roots, you can solve expressions that involve these operations more effectively. Recognize that roots are a central concept in understanding how exponents are structured.

Expression Equivalence

Expression equivalence is all about understanding when two different expressions hold the same value. In terms of powers and roots, this becomes an examination of algebraic relationships between expressions.
In the given exercise, comparing \(a^{1/k}\) and \(\frac{1}{a^k}\) involves exploring whether these two expressions can be identical under any circumstances. **Equivalence** happens when both sides of an equation evaluate to the same number for the same input.
One way to test for equivalence is by substituting values into the expressions and checking whether they yield equal results. However, more robust methods involve using properties of exponents to manipulate and simplify expressions.
Specifically, for \(a^{1/k}\) to equal \(\frac{1}{a^k}\), they would need to simplify to the same basic form or have simplifying conditions, such as \(a = 1\). If no such simplification or consistent value satisfies the condition, the expressions are not equivalent.

Properties of Exponents

The properties of exponents are fundamental rules that help in solving and simplifying algebraic expressions involving powers. Here are some key properties to remember:

• Product of Powers: \(a^m \times a^n = a^{m+n}\) - add exponents when multiplying with the same base.
• Power of a Power: \((a^m)^n = a^{m\cdot n}\) - multiply exponents when raising a power to another power.
• Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\) - subtract exponents when dividing with the same base.
• Zero Exponent: \(a^0 = 1\) for any non-zero \(a\) - any number to the zero power is one.
• Negative Exponent: \(a^{-m} = \frac{1}{a^m}\) - represents reciprocal of the base raised to the positive power.

Understanding these rules allows students to manipulate and simplify expressions efficiently, which is often required when determining the equivalency of expressions like in the provided exercise. In our exercise, these properties guide us in redefining \(a^{1/k} \times a^k\) or \((a^{1/k})^k\) to test for potential simplification.