Simplifying Cube Roots and Exponents: A Comprehensive Guide

Simplifying expressions involving cube roots and exponents can often appear daunting, especially when multiple operations are involved. This article will guide you through the process of simplifying expressions such as (sqrt[3]{x^2}^{frac{3}{4}}). By following a systematic approach, you can simplify such expressions effectively.

Step-by-Step Simplification

To simplify the expression (sqrt[3]{x^2}^{frac{3}{4}}), we can follow the steps outlined below:

Step 1: Rewrite Using Exponents

First, let's express the cube root using fractional exponents.

(sqrt[3]{x^2} x^{frac{2}{3}})

This allows us to rewrite the initial expression as:

(x^{frac{2}{3}}^{frac{3}{4}})

Step 2: Use the Power of a Power Property

When raising a power to another power, you multiply the exponents.

(x^{frac{2}{3} cdot frac{3}{4}} x^{frac{2 cdot 3}{3 cdot 4}} x^{frac{6}{12}} x^{frac{1}{2}})

Step 3: Rewrite the Result in Radical Form

The expression (x^{frac{1}{2}}) can be rewritten in radical form:

(x^{frac{1}{2}} sqrt{x})

Therefore, the simplified form of (sqrt[3]{x^2}^{frac{3}{4}}) is (sqrt{x}).

Additional Examples and Properties

Understanding the properties of exponents and radicals is crucial for simplifying these expressions. Let's consider another example:

Example 1: Simplify the expression {x^2}^{1/4}.

When power is raised over power, they get multiplied:

{{x^2}^{1/4} x^{2cdot frac{1}{4}} x^{frac{1}{2}}

This can be rewritten in radical form:

x^{frac{1}{2}} sqrt{x}

Example 2: Simplify the expression (sqrt[3]{x^2}^{frac{3}{4}}).

Following the same steps as above:

1. Rewrite using exponents: (sqrt[3]{x^2} x^{frac{2}{3}})

2. Apply the power of a power property: (x^{frac{2}{3} cdot frac{3}{4}} x^{frac{2 cdot 3}{3 cdot 4}} x^{frac{6}{12}} x^{frac{1}{2}})

3. Rewrite the result in radical form: (x^{frac{1}{2}} sqrt{x})

Conclusion

Mastering the skills to simplify expressions involving cube roots and exponents opens up a world of mathematical problem-solving. By following the steps outlined in this article, you can confidently tackle such expressions and simplify them efficiently.

For further reading and practice, consider exploring more complex expressions and their simplifications. Understanding the underlying properties will enhance your ability to manipulate these expressions accurately.