- Multiply the exponents inside the parentheses by adding them: \(\frac{4}{3} + \frac{2}{3} = 2\).
- Simplify the expression inside the parentheses to \(x^2\).
- Raise the simplified expression to the power of \(\frac{1}{3}\) by multiplying the exponents: \(2 \times \frac{1}{3} = \frac{2}{3}\).
- Conclude that the expression is equivalent to \(\boxed{x^{\frac{2}{3}}}\).
Explanation
- Analysis of the Expression and Relevant Exponent Laws
To simplify the expression \(\left(x^{\frac{4}{3}} x^{\frac{2}{3}}\right)^{\frac{1}{3}}\), we need to use the fundamental laws of exponents. These laws allow us to handle powers of the same base by combining their exponents through addition or multiplication. The expression we are working with is composed of a product of powers inside parentheses, followed by an overall power applied to the result. We will process this step-by-step.
- Simplifying the Term Inside the Parentheses
First, we focus on the expression inside the parentheses: \(x^{\frac{4}{3}} \cdot x^{\frac{2}{3}}\). According to the product rule for exponents, when multiplying two powers with the same base, we add the exponents: \(x^a \cdot x^b = x^{a+b}\). Applying this rule here, we have:
\[x^{\frac{4}{3}} \cdot x^{\frac{2}{3}} = x^{\frac{4}{3} + \frac{2}{3}} = x^{\frac{6}{3}} = x^2\]
Thus, the entire expression simplifies to \(\left(x^2\right)^{\frac{1}{3}}\).
- Applying the Power of a Power Rule
Now we have the expression in the form \(\left(x^2\right)^{\frac{1}{3}}\). We use the power of a power rule for exponents, which states that when raising a power to another exponent, we multiply the exponents: \(\left(x^m\right)^n = x^{m \cdot n}\). Applying this rule to our expression, we get:
\[\left(x^2\right)^{\frac{1}{3}} = x^{2 \cdot \frac{1}{3}} = x^{\frac{2}{3}}\]
This matches the second option provided in the problem.
- Final Conclusion and Verification
Comparing our result \(x^{\frac{2}{3}}\) with the given choices:
- \(x^{\frac{2}{9}}\)
- \(x^{\frac{2}{3}}\)
- \(x^{\frac{8}{27}}\)
- \(x^{\frac{7}{3}}\)
We confirm that the equivalent expression is \(x^{\frac{2}{3}}\). The final answer is \(\boxed{x^{\frac{2}{3}}}\).
Examples
Understanding how to manipulate exponents is essential in many fields, such as computer science when calculating the complexity of algorithms or in physics when working with scientific notation and dimensional analysis. For example, if a data storage system triples its capacity every time a certain process is performed twice, you might use fractional exponents to calculate the total capacity increase over a single fractional step of that process. Mastering these rules ensures you can simplify complex formulas into manageable forms, saving time and reducing the risk of errors in your calculations.