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Chapter 7: Problem 6

Simplify. \(\sqrt[3]{\frac{2}{3 x}}\)

Short Answer

Expert verified
\( \frac{\sqrt[3]{2}}{\sqrt[3]{3} \cdot \sqrt[3]{x}} \)

Step by step solution

01

Understand the Expression

The expression given is \( \sqrt[3]{\frac{2}{3x}} \). This is a cube root of a fraction. To simplify it, you can separate the cube root over the numerator and the denominator based on the property of radicals: \( \sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}} \).
02

Apply Cube Root Separately

Apply the cube root to both the numerator and the denominator: \( \frac{\sqrt[3]{2}}{\sqrt[3]{3x}} \). Here, \( \sqrt[3]{2} \) remains as it is, and for the denominator, you have \( \sqrt[3]{3x} = \sqrt[3]{3} \cdot \sqrt[3]{x} \).
03

Simplify Expression (if necessary)

The expression is now \( \frac{\sqrt[3]{2}}{\sqrt[3]{3} \cdot \sqrt[3]{x}} \). If any further simplification is possible or required, it would depend on the context or additional constraints. In this basic case, the expression is simplified as is.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cube Roots
Cube roots are a way of finding which number, when multiplied by itself three times, gives us the original number. They are similar to square roots, but instead of two numbers multiplied together, it's three.
For instance, the cube root of 8 is 2, because \(2 \times 2 \times 2 = 8\). In notation, we express cube roots using the symbol \(\sqrt[3]{}\), such as \(\sqrt[3]{8} = 2\).
Cube roots work the same way for both whole numbers and fractions, and knowing how to handle them is essential in simplifying expressions.
Properties of Radicals
When dealing with radicals, there are some handy properties that make simplifying expressions much easier. One important property is the ability to "distribute" a radical over a fraction.
  • The property says: \(\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}\), where "n" is the degree of the root.
  • This means, in our example \(\sqrt[3]{\frac{2}{3x}}\), you can separate the radicals to simplify calculations to \(\frac{\sqrt[3]{2}}{\sqrt[3]{3x}}\).
This separation makes it easier to manipulate and simplify the expression further, especially when dealing with cube roots or higher-level radicals.
Simplifying Fractions Involving Radicals
Simplifying fractions that involve radicals may look intimidating, but once broken down, it can be straightforward. The goal is to express the fraction in its simplest form by applying different properties of radicals.
The expression \(\sqrt[3]{\frac{2}{3x}}\) separates into \(\frac{\sqrt[3]{2}}{\sqrt[3]{3} \cdot \sqrt[3]{x}}\).
When simplifying such expressions, look at each component:
  • Determine if further simplification of individual radicals is possible. For example, check if there is a cube term, like \(8\) which can further be simplified to \(2\).
  • Examine the variables. In the expression \(\sqrt[3]{3x}\), \(\sqrt[3]{x}\) remains as is if \(x\) doesn't simplify to a whole number.
Using these strategies helps you handle and express fractions with radicals in their simplest possible form.

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Most popular questions from this chapter

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