Which expression is equivalent to $x^{-\frac{5}{3}}$ ?
To find an expression equivalent to \(x^{-\frac{5}{3}}\), we apply the rules of exponents systematically. We start by using the negative exponent rule, which states that for any non-zero real number \(x\) and positive rational number \(n\), \(x^{-n} = \frac{1}{x^n}\). Applying this to our expression, we get:
\[x^{-\frac{5}{3}} = \frac{1}{x^{\frac{5}{3}}}\]
Next, we focus on the denominator \(x^{\frac{5}{3}}\). We use the rule for rational exponents, which states that \(x^{\frac{m}{n}} = \sqrt[n]{x^m}\). In our expression, \(m = 5\) and \(n = 3\). Substituting these values into the rule, we can rewrite the denominator as:
\[x^{\frac{5}{3}} = \sqrt[3]{x^5}\]
Now, we combine the results from the previous steps. By substituting the equivalent form of the denominator back into our initial fraction, we find:
\[x^{-\frac{5}{3}} = \frac{1}{x^{\frac{5}{3}}} = \frac{1}{\sqrt[3]{x^5}}\]
Comparing this result to the given options, we identify that the correct equivalent expression is \(\frac{1}{\sqrt[3]{x^5}}\).
Understanding how to convert between exponential and radical forms is essential in various fields, such as physics and engineering, where formulas involving power laws often appear. For example, in calculating the period of a planet's orbit using Kepler's Third Law (where the square of the orbital period is proportional to the cube of the semi-major axis), you might encounter expressions with rational exponents. Being able to simplify and rewrite these expressions allows you to better interpret the mathematical relationships underlying the physical phenomena you are studying.