Problem 3

Explain why when different powers of the same quantity are multiplied together their exponents are added.

Solution. Consider xm and xn. By definition,

xm=x⋅x⋯x (multiply x by itself m times)

and

xn=x⋅x⋯x(multiply x by itself n times)

Their product is x multiplied by itself m times multiplied by x multiplied by itself n . Another words, x multiplied by itself m+n times

⇒ xm⋅xn=x(m+n)

An alternative explanation uses logarithms:

log(xm)=m*log(x)

and

log(xn)=n*log(x)

⇒log(xm⋅xn)=(m+n)*log(x)

and exponentiating both sides gives the same result.

Answer: xm⋅xn=x(m+n)