Problem 37 - Algebra 2 and Trigonometry

Question

In \(11-38,\) evaluate each expression in the set of real numbers. $$ \sqrt[k]{1}, k \text { an integer greater than } 2 $$

Step-by-Step Solution

Answer

The expression \( \sqrt[k]{1} \) evaluates to 1 for any integer \( k > 2 \).

Step 1: Understand the Problem

The expression \( \sqrt[k]{1} \) represents the \( k \)-th root of 1, where \( k \) is an integer greater than 2. We need to evaluate this expression within the set of real numbers.

Step 2: Recall the Property of Roots

Recall that for any real number \( a \), \( \sqrt[k]{a} = a^{1/k} \). In this case, \( a = 1 \), so \( \sqrt[k]{1} = 1^{1/k} \).

Step 3: Evaluate 1 to Any Power

Any real number 1 raised to any real number power is 1, i.e., \( 1^{1/k} = 1 \) for any integer \( k > 0 \). This remains true even when \( k > 2 \).

Step 4: Conclude the Solution

Since \( 1^{1/k} = 1 \) for any positive integer \( k \), the expression \( \sqrt[k]{1} \) simplifies to 1 for all integer \( k > 2 \).

Key Concepts

Real Numbers

Real numbers are a fundamental part of mathematics and a key concept when evaluating expressions like the one given in our problem. These numbers include all the numbers you can think of, both positive and negative, that can be placed on a continuous number line. They encompass:

• Natural numbers (like 1, 2, 3),
• Whole numbers (including 0),
• Integers (both positive and negative whole numbers),
• Rational numbers (which can be expressed as a fraction like 3/4 or 1.25), and
• Irrational numbers (like \( \pi \) and \( \sqrt{2} \) that cannot be represented as a simple fraction).

Real numbers are essential when performing mathematical operations because they enable us to solve a wide variety of equations and expressions accurately within their scope. When we take the k-th root of 1, we are doing so within this set, ensuring that our answer aligns with the established properties of real numbers.

Power Properties

Understanding power properties is crucial for evaluating expressions involving k-th roots. These properties help simplify complex operations involving exponents. Some key power properties include:

• Power of a Product: \( (ab)^n = a^n \times b^n \).
• Power of a Power: \( (a^m)^n = a^{m \cdot n} \).
• Power of a Quotient: \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \) (assuming \( b eq 0 \)).

In the context of k-th roots, we use the fact that \( a^{1/k} \) is the same as taking the k-th root of \( a \). For the number 1, this simplifies dramatically, because \( 1^n \) is always 1 for any integer \( n \). Thus, \( 1^{1/k} \) is still 1, reaffirming that the k-th root of 1 within the set of real numbers remains 1.

Evaluating Expressions

Evaluating expressions requires step-by-step analysis and a good grasp of mathematical properties. Let's break down the process using our specific example of \( \sqrt[k]{1} \):

• Identify the Expression: We're finding the k-th root of 1.
• Recall Basic Properties: For any real number \( a \), the expression \( a^{1/k} \) defines the k-th root.
• Apply Power Properties: For our problem, it's key to note that \( 1^{1/k} = 1 \) for any integer \( k > 2 \). This is because the power of 1 remains 1 regardless of the exponent.

This understanding lets us confidently conclude that the evaluated expression of \( \sqrt[k]{1} \) is 1 in any real number context. This methodology showcases the importance of using established mathematical rules to simplify and solve expressions efficiently.