For use in later chapters, we now further develop the use of exponents and radicals.

In previous chapters, we have used only exponents that are integers, and by introducing exponents that are fractions we will show the relationship between exponents and radicals (for example, we will show that $\sqrt{x}\hspace{0.17em}=\hspace{0.17em}{x}^{\mathrm{1/2}}$). In more advanced math and in applications, it is more convenient to use fractional exponents rather than radicals.

As we noted in Chapter 6, the use of symbols led to advances in mathematics and science. As variables became commonly used in the 1600s, it was common to write, for example, $x^3$ as xxx. For larger powers, this is obviously inconvenient, and the modern use of exponents came into use. The first to use exponents consistently was the French mathematician René Descartes in the 1630s.

The meaning of negative and fractional exponents was first found by the English mathematician Wallis in the 1650s, although he did not write them as we do today. In the 1670s, it was the great English mathematician and physicist Isaac Newton who first used all exponents (positive, negative, and fractional) with modern notation. This improvement in notation made the development of many areas of mathematics, particularly calculus, easier. In this way, it led to many advances in the applications of mathematics.

As we develop the various operations with exponents and radicals, we will show their uses in some technical areas of application. They are used in a number of formulas in areas such as electronics, hydrodynamics, optics, solar energy, and machine design.