Problem 108 - College Algebra

Question

Simplify by reducing the index of the radical. $$\sqrt[12]{x^{4} y^{8}}$$

Step-by-Step Solution

Answer

The simplified expression of \( \sqrt[12]{x^{4} y^{8}} \) is \( x^{1/3}y^{2/3} \).

Step 1: Apply the Root-to-Power Conversion Rule

Convert the radicals into exponentiation expressions using the rule \( \sqrt[n]{a^m} = a^{m/n} \). Thus, we have \( x^{4/12} y^{8/12} \).

Step 2: Simplify the Fractional Exponents

Reduce the fractional exponents by dividing the numerator and the denominator by their greatest common divisor. So, \( x^{4/12} \) simplifies to \( x^{1/3} \), and \( y^{8/12} \) simplifies to \( y^{2/3} \).

Step 3: Combine the Expressions

Combine the expressions to form the final simplified expression, yielding \( x^{1/3}y^{2/3} \).