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

Verified step by step guidance

1. Identify the expression inside the radical: \(\sqrt[12]{x^4 y^8}\), which is a 12th root of the product \(x^4 y^8\).
2. Recall that the 12th root of a variable raised to a power can be rewritten using rational exponents: \(\sqrt[12]{x^4 y^8} = (x^4 y^8)^{\frac{1}{12}}\).
3. Apply the exponent to each factor inside the parentheses separately: \((x^4)^{\frac{1}{12}} \cdot (y^8)^{\frac{1}{12}}\).
4. Simplify the exponents by multiplying powers: \(x^{\frac{4}{12}} \cdot y^{\frac{8}{12}}\).
5. Reduce the fractions in the exponents to their simplest form: \(x^{\frac{1}{3}} \cdot y^{\frac{2}{3}}\), which is the simplified expression with reduced radical index.

Key Concepts

Radical Expressions and Indices

A radical expression involves roots, such as square roots or twelfth roots, indicated by an index. The index shows the root degree, for example, the 12th root means raising a number to the power of 1/12. Understanding how to interpret and manipulate these indices is essential for simplifying radicals.

Exponent Rules and Fractional Exponents

Exponents can be expressed as fractions to represent roots, where the denominator is the root index. For example, the 12th root of x^4 can be written as x^(4/12). Applying exponent rules, such as reducing fractions and multiplying exponents, helps simplify expressions with radicals.

Simplifying Radicals by Reducing the Index

Reducing the index of a radical means rewriting the radical with a smaller root index by factoring exponents and extracting powers. This process often involves dividing exponents by the greatest common divisor with the index, simplifying the radical to an equivalent but simpler form.