Question

Simplify. $$\sqrt[6]{\frac{4 x^{6}}{9}}$$

Short answer

\(\frac{\sqrt[6]{4} \cdot x}{\sqrt[6]{9}}\)

Step-by-step solution

Identify the elements of the expression

First, recognize that the expression \(\sqrt[6]{\frac{4 x^{6}}{9}}\) consists of a sixth root and a fraction within the radicand. The expression breaks down into the sixth root of the numerator \(4 x^6\) and the sixth root of the denominator \(9\).

Simplify the radicand

Simplify the radicand by breaking it down into the sixth root of the numerator and the sixth root of the denominator: \(\sqrt[6]{4 x^6}\) over \(\sqrt[6]{9}\). The sixth root of \(x^6\) is \(x\) because \(x^{6/6} = x^1 = x\), and the sixth root of \(4\) remains as is because it cannot be simplified further.

Simplify the root of the fraction

Now consider the denominator \(\sqrt[6]{9}\). Since 9 does not have a perfect sixth root, it remains under the root sign. However, if the numbers were perfect powers of 6, they could be further simplified outside of the root.

Write the simplified form

Write the final simplified form of the expression by combining the simplified numerator and the unchanged denominator: \(\frac{\sqrt[6]{4} \cdot x}{\sqrt[6]{9}}\).

Key concepts

Sixth Root Simplification

Understanding how to simplify the sixth root of a number or an algebraic expression is a crucial skill in algebra. The sixth root of a number, represented by the radical symbol \( \sqrt[6]{...} \) and the number 6 above it, refers to finding a value that when raised to the power of six, produces the original number within the radical. Simplifying such radicals often involves looking for perfect powers of 6 within the radicand—the number or expression inside the radical sign—and separating them from the remaining factors.

For example, when taking the sixth root of \( x^6 \) the result is \( x \) because \( x \) to the power of 6 returns us to our original radicand (\( x^6 \)). This is known as a 'perfect sixth power'. If you encounter terms like \( 4 \) which are not perfect powers of 6, they remain within the radical because they cannot be simplified in this manner.

In essence, the process of simplifying the sixth root involves identifying and extracting perfect sixth powers, while leaving other numbers inside the radical. This can significantly simplify an expression, making it easier to use in further calculations or applications.

Radical Expressions

Radical expressions are mathematical expressions that involve roots, whether square roots, cube roots, or any other roots. They are an integral part of algebra and higher-level mathematics, serving as the bridge to understanding polynomial equations, complex numbers, and even exponential functions.

Consider the sixth root of a ratio like \( \sqrt[6]{\frac{4 x^{6}}{9}} \) from our exercise. Such an expression includes a radical sign indicating the root, a numerator (\( 4x^6 \)), and a denominator (\( 9 \)). Simplifying radical expressions can seem daunting, but approaching them step-by-step makes the process manageable. First, we identify each component of the radical, then consider any potential simplifications separately in the numerator and the denominator. Simplifying radicals often means finding the factor of the radicand that is a power of the index (e.g., the sixth power when dealing with a sixth root) and extracting it outside the radical sign.

Simplify Fractional Exponents

Fractional exponents are another way to express powers and roots and thus are closely related to radical expressions. They provide a convenient notation for simplifying expressions, especially when you're dealing with roots of variables. The fractional exponent consists of a numerator and a denominator; the numerator represents the power, while the denominator corresponds to the root.

For instance, if you have \( x^{\frac{6}{6}} \), this is equivalent to taking the sixth root of \( x^6 \) (the power) and simplifying it down to \( x \). In the realm of fractional exponents, simplification is done by reducing the fraction to its lowest terms or by expressing the exponent form in a radical form and vice versa. We often simplify expressions with fractional exponents by finding common factors in the numerator and denominator and canceling them out.

When simplifying radicals or fractional exponents, it's always a good idea to look for a way to express the expression in its simplest form, which means minimizing the exponent and radical as much as possible without altering the value of the expression.