Mathematics

Problem

Let

p

be a non-zero integer and

m,\,n

be any integers. a. (i) Prove that

p^{m}\,\times\,p^{-n}=p^{\,(m-n)}

using only the fundamental laws of exponents. b. (ii) Using the result from part (i), simplify

7^{6}\,\times\,7^{-9}\,\times\,7^{4}

and write the answer with a single exponent. c. (iii) If

p^{a}\,\times\,p^{b}\,\times\,p^{c}=p^{5}

and

a+b+c=-2

, show that

a+b+c-5=-7

and interpret this result in terms of exponents.