Problem 101 - College Algebra

Question

In Exercises \(95-102,\) simplify by reducing the index of the radical. $$\sqrt[9]{x^{6} y^{3}}$$

Step-by-Step Solution

Answer

The simplified expression of \( \sqrt[9]{x^{6} y^{3}} \) is \(x^{2/3} \cdot y^{1/3}\)

Step 1: Identify Base and Exponent

The base in the numerator of the expression are \(x\) and \(y\). The exponents are 6 and 3, respectively. The index of the radical is 9.

Step 2: Apply the Property of Roots

We know that if \( a, b > 0 \), then \( \sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b} \). The formula simplifies the expression to: \( \sqrt[9]{x^{6}} \cdot \sqrt[9]{y^{3}} \)

Step 3: Simplify Exponents

To simplify the powers, divide each exponent by the index of the root. The general rule is, given \( \sqrt[n]{a^m} \), it simplifies to \( a^{m/n} \). So, \( \sqrt[9]{x^{6}} \) simplifies to \(x^{6/9}\) or \(x^{2/3}\), and \( \sqrt[9]{y^{3}} \) simplifies to \(y^{3/9}\) or \(y^{1/3}\)

Step 4: Final Simplified Expression

The final answer is the product of the simplified exponents from the previous step. This results in \(x^{2/3} \cdot y^{1/3}\)

Key Concepts

Radical Expressions

When you encounter a mathematical expression that includes a root sign, such as a square root, cube root, or any higher root, you are dealing with a radical expression. Simplifying radical expressions involves manipulating these expressions to make them easier to understand and work with.

For example, in the exercise \(\sqrt[9]{x^{6} y^{3}}\), both \(x^{6}\) and \(y^{3}\) are under a ninth root, which makes the entire expression a radical. To simplify this, we look to reduce the exponents while adhering to the root, aiming for a more straightforward expression. By applying properties of roots and rational exponents, radical expressions can be rewritten to seem less daunting and more manageable in mathematical operations.

Properties of Roots

Roots, the inverse operation to raising a number to a power, have their own set of properties that can greatly simplify radical expressions. One such property is the product property of roots, which states that the nth root of a product is equal to the product of the nth roots of the factors. Symbolically, \(\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}\).

This allows us to separate an expression like \(\sqrt[9]{x^{6} y^{3}}\) into \(\sqrt[9]{x^{6}} \cdot \sqrt[9]{y^{3}}\) as seen in the step-by-step solution. Furthermore, these individual roots can be simplified by reducing the exponents based on the index of the root, which is another crucial property useful in many mathematical contexts.

Rational Exponents

The concept of rational exponents is intimately connected to roots. A rational exponent represents both an exponent and a root, which can be understood with the expression \(a^{m/n}\). This denotes the nth root of \(a\) raised to the power of m.

In the exercise provided, we see how the exponent rule is applied: \(\sqrt[9]{x^{6}}\) is rewritten as \(x^{6/9}\), and similarly, \(\sqrt[9]{y^{3}}\) as \(y^{3/9}\). This illustrates how radical expressions can be converted into expressions with rational exponents, which are often easier to manipulate algebraically. Rational exponents provide a powerful tool for simplification and serve as a cornerstone for understanding higher-level mathematics.