Problem 106 - College Algebra Essentials

Question

In Exercises \(101-108\), simplify by reducing the index of the radical. $$\sqrt[9]{x^{6}}$$

Step-by-Step Solution

Answer

The simplified form of the radical \(\sqrt[9]{x^{6}}\) is \(\sqrt[3]{x^{2}}\).

Step 1: Write the radical as a fractional exponent

The first step is to rewrite the radical \(\sqrt[9]{x^{6}}\) as a fractional exponent. The rule here is that the index of the root (9 in this case) becomes the denominator of the fraction and the power under the root (6 in this case) becomes the numerator. Therefore, \(\sqrt[9]{x^{6}}\) can be rewritten as \(x^{6/9}\).

Step 2: Simplify the fractional exponent

Next, we can simplify the fractional exponent \(x^{6/9}\). Both the numerator (6) and the denominator (9) can be divided by their greatest common divisor which is 3. We get \(x^{2/3}\).

Step 3: Rewrite the simplified fractional exponent as a radical

Finally, we rewrite the simplified fractional exponent \(x^{2/3}\) as a radical. This is the inverse of the process we did in step 1. The denominator of the fractional exponent becomes the index of the root, and the numerator becomes the power under the root. Therefore, \(x^{2/3}\) can be rewritten as \(\sqrt[3]{x^{2}}\).

Key Concepts

Simplification of Radicals

When simplifying radicals like \( \sqrt[9]{x^{6}} \), the goal is to convert them into an easier form. Radicals can be complex, but with practice, they become straightforward. A radical expression involves a root, denoted by the symbol \( \sqrt{} \). The number inside this root is the radicand, and the number outside (in this case, 9) is the index.

To simplify, follow these steps:

• Convert the radical: Change the expression to a fractional exponent, making it easier to handle. For instance, \( \sqrt[9]{x^{6}} \) becomes \( x^{6/9} \).
• Reduce the fraction: Divide both the numerator (6) and the denominator (9) by their Greatest Common Divisor (GCD), simplifying the exponent to \( x^{2/3} \).
• Convert back to radical (if necessary): If the problem requires, convert the fractional form back into a radical, yielding \( \sqrt[3]{x^{2}} \).

These steps help make the expressions manageable and lead to a simplified and clearer form.

Fractional Exponents

Fractional exponents present a unique method to express roots. They bridge the gap between traditional exponentiation and taking roots, providing a new viewpoint. For example, the seventh root of a variable \( x \) raised to a power could be expressed as \( x^{m/n} \). Here, the denominator \( n \) indicates the type of root, and the numerator \( m \) signifies the power applied to the radicand.

When handling fractional exponents:

• Understand the notation: \( x^{1/2} \) equals \( \sqrt{x} \), and similarly, \( x^{m/n} \) is equivalent to taking the \( n \)-th root of \( x^{m} \).
• Calculate the GCD: This aids in reducing the fraction \( m/n \) to its simplest form, key to further simplifications.
• Simplify the process: Leveraging fractional exponents can simplify complex problems, allowing seamless conversion between radical and exponential notations.

Grasping this relationship is pivotal for fluid transitions between the forms.

Greatest Common Divisor (GCD)

To simplify expressions like \( x^{6/9} \), finding the Greatest Common Divisor (GCD) is crucial. The GCD is the largest number that divides two or more integers without leaving a remainder. In our example, with numbers 6 and 9, the GCD is 3.

Steps for finding the GCD can be simple:

• Prime Factorization: Break each number into its prime factors. For instance, \( 6 = 2 \times 3 \) and \( 9 = 3 \times 3 \).
• Identify common factors: Look for shared prime factors in both numbers. The common prime factor here is 3.
• Calculate the GCD: Multiply the shared prime factors. Here, from 3 in both factor lists, giving us a GCD of 3.

Utilizing the GCD simplifies the fraction \( 6/9 \) to \( 2/3 \), rendering the expression \( x^{6/9} \) more concise as \( x^{2/3} \). Recognizing the GCD's role significantly eases simplification tasks.