There are three basic things to remember when working with exponents. All equations involving exponents can, in essence, be worked out with the basic memorization of the following three rules:
1) x^n = (x)(x)...(x), where n represents a specific number of factors of x to be multiplied.
2) x^-n = 1/x^n, as long as x does not equal 0.
3) x^1/n = x to the nth root; for example, 4^(1/2) is the square root of 4.
Modern-day algebra is derived from the Arabic al-jabr, developed in part by Abu Jafar Muhammed ibn Musa al-Khwarizmi in the early ninth century. Al-Khwarizmi is credited with bringing what are now known as Arabic numerals -- the numerals 0 through 9 as we know them -- to western Europe, and consequently, the entire Western world. His most important work was the book "al-Kitab al-mukhtasar fi hisab al-jabr w'al-muqabala," or "The Compendious Book on Calculation by Completion and Balancing." The book was written entirely in prose, with none of the symbols we have since come to associate with algebra, and explained how to solve six varieties of quadratic and linear equations.
Two hundred years after al-Khwarizmi's book was published, mathematician Abu Bakr al-Karaji continued his predecessor's work and became the first person to name the powers in the x^n format, making it possible to work with polynomials and develop laws regarding their addition, subtraction, division and multiplication. His work paved the way for later Western mathematicians to further develop the laws of exponents. In the mid-12th century, another Arabic mathematician, Al-Samaw’al bin Yahya bin Yahuda al-Maghribi, further developed the theory by introducing negative coefficients and early work with using exponents to divide polynomials.
Five hundred years after the work of the earlier Arabic mathematicians mentioned, René Descartes modernized the algebraic formula for exponents into what we recognize today. It was Descartes who originally developed the notation of exponents as x^n. He also developed the Descartes Rule of Signs, which is used to find the positive and negative roots of equations.