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Chapter 0: Problem 108

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

Short Answer

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

Step by step solution

01

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 \( x^{4/12} y^{8/12} \).
02

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 \( y^{2/3} \).
03

Combine the Expressions

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

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