Problem 22 - College Algebra with Modeling and Visualization

Question

Use positive exponents to rewrite. $$ \sqrt[5]{x^{2}} $$

Step-by-Step Solution

Answer

\( x^{2/5} \)

Step 1: Understand the Problem

The problem asks us to rewrite the expression \( \sqrt[5]{x^{2}} \) using positive exponents. This involves converting roots into a fractional exponent form.

Step 2: Use the Rule of Radicals and Exponents

The general form of expressing a root as a fractional exponent is \( \sqrt[n]{x^{m}} = x^{m/n} \). In this expression, the index \( n \) of the root becomes the denominator of the exponent, and the exponent \( m \) of the radicand (the number inside the root) becomes the numerator.

Step 3: Apply the Rule to the Given Problem

Apply the rule for converting roots to exponents to the given expression \( \sqrt[5]{x^{2}} \). Here, \( n = 5 \) and \( m = 2 \). Thus, the expression \( \sqrt[5]{x^{2}} = x^{2/5} \).

Step 4: Simplification and Conclusion

Since \( x^{2/5} \) already has a positive exponent, the expression is now properly rewritten using positive exponents.

Key Concepts

Fractional Exponents

When we talk about fractional exponents, we refer to a way of expressing roots with exponents rather than the typical radical sign. It's an essential concept in algebra because it allows us to rewrite and manipulate expressions more flexibly. Understanding fractional exponents is pretty straightforward. Consider the expression \( x^{m/n} \):

• The numerator \( m \) signifies the power to which the number \( x \) is raised.
• The denominator \( n \) represents the root you're taking of that power.

This means that \( x^{m/n} \) is equivalent to taking the \( n \)-th root of \( x^m \). For example, \( x^{2/5} \) is the fifth root of \( x^2 \). This conversion can make complex computations simpler and more manageable. Learning to switch between these forms is valuable, enabling algebraic prowess with different kinds of equations.

Radicals

Radicals are expressions that use the root symbol, known as the radical sign. They are generally written as \( \sqrt[n]{x} \) where \( n \) is the index, and \( x \) is the radicand. When no index is specified, it's understood to be a square root. Radicals are connected to fractional exponents as they can be transformed into such exponents easily:

• The expression \( \sqrt[3]{x} \) can be rewritten as \( x^{1/3} \).
• For \( \sqrt[5]{x^2} \), it becomes \( x^{2/5} \).

Radicals are vital in various math categories because roots are needed whenever we solve equations for unknown basics, such as quadratic equations or higher-degree polynomials. Converting between radicals and fractional exponents is important because it often makes solving and simplifying problems much simpler.

Converting Expressions

The process of converting expressions involves changing mathematical forms from one notation to another while keeping the same value. This often involves switching between radicals and fractional exponents. Understanding this process is crucial for simplifying expressions and solving equations efficiently. Here’s a general step-by-step approach:

• Identify the radical in your expression. Determine the index \( n \) and the power \( m \), if any.
• Rewrite the expression as a fractional exponent: \( \sqrt[n]{x^m} = x^{m/n} \).
• Simplify if possible, maintaining positive exponents.

For instance, the expression \( \sqrt[5]{x^2} \) is converted to \( x^{2/5} \) using these steps. By converting to fractional exponents, you streamline the resolution of complex algebraic expressions and uncover new ways to manipulate and understand them. Simplifying complex expressions becomes easier, and solving problems more intuitive.