Problem 9 - Elementary and Intermediate Algebra

Question

Assume for all exercises that even roots are of non- negative quantities and that all denominators are nonzero. Write an equivalent expression using radical notation and, if possible, simplify. $$ x^{1 / 6} $$

Step-by-Step Solution

Answer

\( \sqrt[6]{x} \)

Step 1: Identify the Exponent

The given expression is \(x^{1/6}\). Notice that the exponent is a fraction where the numerator is 1 and the denominator is 6.

Step 2: Rewrite Using Radical Notation

To express \(x^{1/6}\) using radical notation, remember that \(x^{1/n}\) is equivalent to the nth root of x. So, \(x^{1/6}\) can be written as \( \sqrt[6]{x} \).

Step 3: Check for Simplification

Next, check if \( \sqrt[6]{x} \) can be simplified further. Since \(x\) is already under the 6th root, it is in its simplest form.

Key Concepts

Fractional Exponents

Fractional exponents might seem intimidating at first, but they are just another way to represent roots of numbers. They help simplify and manipulate expressions more easily. A fractional exponent, such as in the expression \(x^{1/6}\), consists of a numerator and a denominator. The numerator represents the power, while the denominator shows the root. For instance, \[ x^{a/b} \] can be interpreted as the \((b)\)-th root of \(x\) raised to the power of \((a)\). This relation is crucial for converting between radical and exponential forms.
Examples:

• \( x^{1/2} \) can be written as \(\frac{1}{2}\) or \(\sqrt{x}\), which is the square root of \(x\).
• \( x^{3/4} \) translates to \(\sqrt[4]{ x^3}\), implying the fourth root of \(x\) cubed.

Understanding how these conversions work will make solving more complex problems easier and faster. Practice rewriting various expressions to become comfortable with both forms.

Simplifying Expressions

Simplifying mathematical expressions ensures they are as straightforward as possible. This step-by-step approach not only makes calculations easier but also helps in recognizing patterns and important properties. For example, simplifying the expression \( \sqrt[6]{x} \) means realizing that this form cannot be broken down further if \(x\) is just a variable.
Tips for simplifying:

• Always check if the base, coefficients, or exponents can be reduced.
• Use properties of exponents, like \( a^m / a^n = a^{m-n} \) and \((a^m)^n = a^{m*n} \).
• Convert between radicals and exponents to see if it reveals an easier path to simplification.

By diligently applying these rules, the problems become less cumbersome and vastly more manageable. For \( x^{1 / 6} \), there aren't any further simplifications, demonstrating that sometimes the expression provided is already in its simplest form.

Roots of Numbers

Roots are the inverse operations of exponents. Taking a root of a number undoes the process of raising that number to a power. The most commonly encountered roots are square roots \( \sqrt{x} \) and cube roots \( \sqrt[3]{x} \). However, roots can have any positive integer as their index.
Key points about roots:

• The n-th root of \(x\), written as \( \sqrt[n]{x} \), answers the question: what number, when raised to the power of \(n\), returns \(x? \)
• Roots apply to both positive and negative numbers, but even roots (like square roots) of negative numbers result in imaginary numbers.
• Roots can simplify products and quotients just like exponents.

Applying these points to our example: \( \sqrt[6]{x} \)is the sixth root of \(x = x^{1/6} \). No further simplification was needed as the expression reflects the sixth root in its simplest form. Understanding roots helps us rewrite and simplify exponential forms like \( x^{1/6} \) with ease.