- Decompose the fractional exponent \(\frac{10}{3}\) into the sum \(3 + \frac{1}{3}\).
- Use the rule \(x^{a+b} = x^a \cdot x^b\) to rewrite \(x^{\frac{10}{3}}\) as \(x^3 \cdot x^{\frac{1}{3}}\).
- Express the remaining term \(x^{\frac{1}{3}}\) as the cube root \(\sqrt[3]{x}\).
- Conclude that the expression is \(\boxed{\left(x^3\right) \sqrt[3]{x}}\).
Explanation
- Analysis of the Given Expression
The expression given is \(x^{\frac{10}{3}}\). Our goal is to rewrite this expression into an equivalent form by using the properties of exponents and radicals. Specifically, we want to express \(x^{\frac{10}{3}}\) in a way that simplifies the fractional exponent into an integer component and a remaining fractional component, or directly into a radical expression, and compare it with the provided choices. The primary laws of exponents that we will use are \(x^{a+b} = x^a \cdot x^b\) and \(x^{1/n} = \sqrt[n]{x}\).
- Simplifying the Exponent
To simplify \(x^{\frac{10}{3}}\), we look at the exponent \(\frac{10}{3}\). We can divide 10 by 3 to separate the integer part and the fractional part: \(\frac{10}{3} = \frac{9+1}{3} = \frac{9}{3} + \frac{1}{3} = 3 + \frac{1}{3}\). Now, we apply the exponent law \(x^{a+b} = x^a \cdot x^b\):
\[x^{\frac{10}{3}} = x^{3 + \frac{1}{3}} = x^3 \cdot x^{\frac{1}{3}}.\]
Next, we convert the fractional part \(x^{\frac{1}{3}}\) into radical form. By the definition of radicals, \(x^{\frac{1}{n}} = \sqrt[n]{x}\), so \(x^{\frac{1}{3}} = \sqrt[3]{x}\). Substituting this back into our expression, we get:
\[x^3 \cdot \sqrt[3]{x}.\]
- Conclusion
Comparing our result \(x^3 \sqrt[3]{x}\) to the provided options:
- \(\left(x^3\right) \sqrt[3]{x}\) - This matches our simplified expression.
- \(\left(\frac{1}{\sqrt[3]{x}}\right)^{10} = (x^{-1/3})^{10} = x^{-10/3}\) - This is not equal.
- \(\sqrt[3]{x^7} = x^{7/3}\) - This is not equal.
- \(\left(x^3\right) \sqrt{x} = x^3 \cdot x^{1/2} = x^{3.5}\) - This is not equal.
Therefore, the correct expression is \(\left(x^3\right) \sqrt[3]{x}\).
Examples
Exponents and radicals are essential tools in fields like engineering and physics. For instance, when calculating the growth of a population or the decay of radioactive material over time, you often encounter fractional exponents that need to be simplified into radical forms to better understand the rate of change or to perform further calculations more easily. Mastering these algebraic manipulations ensures you can handle complex scientific formulas with confidence.