Final answer:
The question involves simplifying algebraic expressions using exponent rules and algebraic manipulation, which are important skills in high school mathematics.
Step-by-step explanation:
The question requires simplification of algebraic expressions, which involves rules such as exponent addition and division by a term with a negative exponent. The given snippets suggest working through various algebraic processes such as completing the square, manipulating equations, and utilizing exponent rules.
For example, the expression x^p x^q = x^(p+q) is a result of the rule that when you multiply expressions with the same base, you add the exponents. Similarly, x^(-n) = 1/x^n demonstrates how negative exponents denote reciprocals. When simplifying expressions with negative exponents, it is equivalent to dividing by the term with the exponent moved to the denominator as a positive exponent.
Additionally, finding the values of p and q would involve algebraic techniques such as rearranging terms and solving equations, which are essential skills in high school mathematics.