Problem 24 - A Graphical Approach to College Algebra

Question

Use positive rational exponents to rewrite each expression. Assume variables represent positive numbers. $$\sqrt[5]{x^{2}}$$

Step-by-Step Solution

Answer

\(\sqrt[5]{x^{2}} = x^{2/5}\)

Step 1: Understand the Problem

We need to express the fifth root of a variable raised to a power using positive rational exponents.

Step 2: Apply the Definition of Rational Exponents

Recall the definition of a rational exponent: \(a^{m/n} = \sqrt[n]{a^m}\). This means the fifth root of \(x^{2}\) can be expressed as \(x^{2/5}\).

Step 3: Rewrite the Expression

Replace \(\sqrt[5]{x^{2}}\) with \(x^{2/5}\). This is our expression using positive rational exponents.

Key Concepts

Understanding Positive Rational Exponents

When dealing with exponents, we can encounter different types, such as whole numbers, fractions, and negatives. Positive rational exponents are numbers expressed as fractions where both the numerator and the denominator are positive integers. These can be viewed as a blend of "root" and "power" functions in mathematics.

For example, you might see something like this: \(x^{2/5}\). The term "rational" refers to the fact that the exponent is a fraction. In this case, it tells us to take the fifth root of \(x\) and then square it, because the numerator is 2 and the denominator is 5.

• The denominator (5 in this instance) tells us the root, in this case, the fifth root.
• The numerator (2 in this example) instructs us to square the outcome of the root.

Converting between radical and exponent forms is crucial since it helps simplify expressions and solve equations more fluidly. With practice, this becomes second nature.

Techniques for Rewriting Expressions

Rewriting expressions using rational exponents instead of roots makes calculations and algebraic manipulation more manageable. When you see \(\sqrt[n]{x^m}\), you can directly translate this into \(x^{m/n}\). This simplification allows for the application of rules of exponents, aiding in easier computations and derivations.

Good practice here is to start by identifying what the root operation is asking you to do. With our example, \(\sqrt[5]{x^{2}}\), you know right away that you’re dealing with the fifth root and a squared variable. Convert this to \(x^{2/5}\) by following the rational exponent rule.

• Conversion helps integrate this into broader algebraic operations, like multiplication and division.
• Rewritten expressions adhere to standard exponent rules \((a^{m} \cdot a^{n} = a^{m+n})\), simplifying complex operations.

With time, rewriting becomes intuitive and boosts your efficiency in handling algebraic tasks.

Applying Variables in Algebra

Variables are symbols used to represent unknown quantities or values in algebraic expressions and equations. Usually denoted by letters such as \(x\), \(y\), or \(z\), they play a crucial role in generalizing mathematical problems to solve a wide range of issues.

In the expression \(\sqrt[5]{x^{2}}\), \(x\) is a variable. This means that \(x\) can be any positive number, giving our expression flexibility and applicability in different scenarios. When we express this using a rational exponent (\(x^{2/5}\)), we rely on the stability of rules governing exponents. Here:

• Variables allow for general solutions across ranges of numbers, making algebra a powerful tool in both theory and application.
• Understanding how to manipulate them with exponents expands the capability to effectively solve and interpret equations in varied contexts.

Ensuring variables represent positive numbers, as in this exercise, prevents issues such as attempts to take imaginary roots, maintaining ease of calculations and relevance to most real-world situations.