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

Verified step by step guidance

1. Identify the expression inside the radical: \(\sqrt[9]{x^6 y^3}\), where the index of the radical is 9.
2. Recall that the radical \(\sqrt[n]{a^m}\) can be rewritten as an exponent: \(a^{\frac{m}{n}}\). So rewrite the expression as \(x^{\frac{6}{9}} y^{\frac{3}{9}}\).
3. Simplify the fractional exponents by reducing the fractions: \(\frac{6}{9} = \frac{2}{3}\) and \(\frac{3}{9} = \frac{1}{3}\), so the expression becomes \(x^{\frac{2}{3}} y^{\frac{1}{3}}\).
4. Rewrite the fractional exponents back into radicals if desired: \(x^{\frac{2}{3}} = \left(\sqrt[3]{x}\right)^2\) and \(y^{\frac{1}{3}} = \sqrt[3]{y}\).
5. Combine the terms under a single cube root: \(\left(\sqrt[3]{x}\right)^2 \cdot \sqrt[3]{y} = \sqrt[3]{x^2} \cdot \sqrt[3]{y} = \sqrt[3]{x^2 y}\).

Key Concepts

Radical Expressions and Indices

A radical expression involves roots, such as square roots or nth roots, where the index indicates the degree of the root. For example, the 9th root (index 9) of a number means finding a value that, when raised to the 9th power, equals the original number. Understanding how to interpret and manipulate these indices is essential for simplifying radicals.

Reducing the Index of a Radical

Reducing the index of a radical means rewriting the radical with a smaller root index by extracting powers from under the radical. This involves expressing the radicand as powers that match or exceed the root index, allowing part of the expression to be taken outside the radical. This process simplifies the expression and makes it easier to work with.

Properties of Exponents and Radicals

The properties of exponents, such as the power of a power and product rules, are crucial when simplifying radicals. For example, the nth root of x^m can be rewritten as x^(m/n). Applying these properties helps convert radicals into fractional exponents, facilitating index reduction and simplification of expressions involving variables and powers.