In calculus it is frequently important to write an expression in the form \( \alpha^{n} \), where \( c \) is a constant and \( n \) is a rational number. Write the following expression in this form.

To rewrite the expression \( \sqrt[3]{x^4} \) in the form \( \alpha^{n} \), we can first express the cube root as a power: \( \sqrt[3]{x^4} = (x^4)^{1/3} \). Then, applying the property of exponents that states \( (a^m)^n = a^{mn} \), we have \( (x^4)^{1/3} = x^{4/3} \). Thus, the expression in the desired form is \( x^{\frac{4}{3}} \). Now, why not explore how exponents work in real life? Imagine you're coding a game where you need to make calculations for character abilities. You might find that using rational exponents makes your equations simpler and more intuitive for scaling up a character's stats, like strength or speed, based on different levels. Additionally, watch out for the common mistake of misapplying the properties of exponents! For instance, remember not to confuse \( (x^m)^n \) with \( x^{mn} \) and also ensure you’re careful with the signs of your exponents when dealing with negative bases. This can lead to some unexpected results that might throw off your calculations!