Problem 13 - A Graphical Approach to College Algebra

Question

Write each expression in radical form. Assume that all variables represent positive real numbers. $$\sqrt[5]{k^{2}}$$

Step-by-Step Solution

Answer

Write as \( \sqrt[5]{k^{2}} \), which is the fifth root form of \( k^2. \)

Step 1: Understand Radicals and Exponents

The expression \( \sqrt[5]{k^{2}} \) can be written in radical form or in exponential form. In radical form, it's already given as a fifth root. We'll understand that radicals are expressions with roots, and \( \sqrt[n]{x} = x^{1/n} \). Here, \( n = 5 \). So, \( \sqrt[5]{k^{2}} = (k^{2})^{1/5} \).

Step 2: Convert to Exponential Form

We can express the radical \( \sqrt[5]{k^{2}} \) using exponents. The expression \( \sqrt[5]{k^{2}} \) means the fifth root of \( k^{2} \). This is equivalent to raising \( k^{2} \) to the power of \( \frac{1}{5} \), giving us the exponential form \( (k^{2})^{1/5} \).

Step 3: Simplify the Expression

Next, apply the property of exponents \((a^m)^n = a^{m \cdot n}\) to the expression \( (k^{2})^{1/5} \). Multiply the exponents: \( 2 \times \frac{1}{5} = \frac{2}{5} \). Thus, the expression simplifies to \( k^{2/5} \ ext{in radical form.} \)

Step 4: Write in Simplified Radical Form

The simplified radical form of \( k^{2/5} \) is \( \sqrt[5]{k^{2}} \), showing the same relation between roots and exponents. Here \( k \) is raised to the power \( 2 \) and then the fifth root is applied.

Key Concepts

Fifth Root

When you first encounter the term "fifth root," it may seem mysterious. However, it is quite straightforward. The fifth root of a number is the value that, when multiplied by itself five times, results in the original number. For example, if we take the fifth root of 32, we are looking for a number that, when raised to the power of five, equals 32. In this case, that number is 2, because 2 raised to the power of 5 (written as \( 2^5 \)) equals 32.

In mathematical notation, a fifth root is expressed as \( \sqrt[5]{x} \). This indicates that the number \( x \) is being operated on by a fifth root, essentially finding a value that satisfies the equation \( x = a^5 \) for some \( a \). Understanding roots, including the fifth root, is crucial for working with both exponents and radicals effectively.

Exponential Form

Exponential form is a concise way of expressing numbers using powers or exponents. It involves expressing a radical expression as a power of a base number. In our exercise, the expression \( \sqrt[5]{k^2} \) can also be written in exponential form. This means expressing the radical as \( (k^2)^{1/5} \).

Understanding how to convert from a radical to an exponential form is essential because it often simplifies calculations. In the expression \( (k^2)^{1/5} \):

• The base is \( k \).
• The exponent is \( \frac{1}{5} \), indicating the fifth root.

Each change between these forms offers a different perspective on the mathematical behavior of the expression, showing how roots relate to fractional exponents.

Exponent Properties

Understanding exponent properties can make working with exponents more accessible and manageable. These properties allow us to simplify and manipulate expressions efficiently. Consider the expression \( (k^2)^{1/5} \). To simplify it, we apply the exponent property \((a^m)^n = a^{m \cdot n}\).

For our expression:

• \( m = 2 \) and \( n = \frac{1}{5} \).
• Multiply these values to get the new exponent: \( 2 \times \frac{1}{5} = \frac{2}{5} \).

This results in the expression \( k^{2/5} \). These properties help not only simplify expressions but also provide a powerful method for solving equations and understanding the relationships between powers. Practicing these properties can aid in seamlessly shifting between different mathematical representations.