Problem 107 - College Algebra

Question

Simplify by reducing the index of the radical. $$\sqrt[9]{x^{6} y^{3}}$$

Step-by-Step Solution

Answer

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

Step 1: Identify the powers and the radical index

The expression provided to us is \( \sqrt[9]{x^{6} y^{3}} \). Here, the index of the radical (ninth root) is 9. The power of x is 6 and that of y is 3.

Step 2: Simplification

From the properties of exponents, we know when a power is raised to another power, the powers are multiplied. So we divide the powers by the index of the radical. Hence, x power becomes \(6 / 9 = 2 / 3\) and, y power becomes \(3 / 9 = 1 / 3\).

Step 3: Final Solution

Now, rewrite the given expression in simplified form by substituting the new powers of x and y back into the original expression. The simplified expression is \(x^{2 / 3} y^{1 / 3}\).