##### 22.2 Exponents: Division rule ${a^x \over a^y}=a^{(x-y)}$

From earlier grade, we were introduced exponents. We already know that in the expression $34^{23}$, 23 is the exponent and 34 is the base. In this chapter, we will use our knowledge of the exponents to better understand Exponential Functions. As the word suggests, Exponential functions are functions that have exponents such as $f(x) = x^2 + 1$ or $g(x) = 3x^5 - 6$.

In this chapter we will learn how to simplify and combine Exponential functions through the. One of the rules we would be talking about is the product rule which states that if you have an expression, $(a^x)(a^y)=a^{(x+y)}$ where the base is the same but the exponents are different. In order for you to simplify this expression you would need to add the exponents.

The second rule would be the division rule which is applicable to Exponential functions that come in the form of a fraction. We know that when an expression is in fraction form, this means that we need to divide the numerator from the denominator, so for any expression $a^x/a^y$, the solution would be $a^{(x-y)}$.

The third rule is the power rule which states that for any expression $(a^x)^y$, you will need to multiply the x and y to get the final exponent. The solution for this expression would be $a^{xy}$.

The other rules would be about the negative exponents, zero exponents and the rational exponents. For the negative exponent rules, such as in the case of $a^{-x}$, to simplify this you would need to get the reciprocal of the expression, 1/a. For the zero exponent rule, any number raised to zero will be equal to 1, and for the rational expression rule, for every expression $a^{1/x}$, the solution would be ${^x}\sqrt{a}$.

Each part of this chapter will give us example of the application of the six rules. At the end of the chapter we are expected to have a more deeper understanding on the rules of exponents as we will apply them to the future chapters.