How can you memorize exponent rules?

1 Answer
Mar 30, 2016

See explanation...

Explanation:

Start with positive integer exponents:

#a^n = overbrace(a xx a xx ... xx a)^"n times"#

Then you can see:

#a^m xx a^n = overbrace(a xx a xx ... xx a)^"m times" xx overbrace(a xx a xx ... xx a)^"n times"#

#=overbrace(a xx a xx ... xx a)^"m + n times" = a^(m+n)#

This is useful when you multiply two numbers that are expressed in scientific notation. For example:

#(1.2 xx 10^3) xx (2.4 xx 10^6)#

#=(1.2 xx 2.4) xx (10^3 xx 10^6)#

#=2.88 xx 10^(3+6)#

#=2.88 xx 10^9#

#color(white)()#
For negative exponents, first note that if #a != 0#:

#a^(-n) = 1/underbrace(a xx a xx ... xx a)_"n times"#

and we find:

#a^n xx a^(-n) = overbrace(a xx a xx ... xx a)^"n times" xx 1/underbrace(a xx a xx ... xx a)_"n times"#

#=overbrace(a xx a xx ... xx a)^"n times"/underbrace(a xx a xx ... xx a)_"n times" = 1#

We find that the rule: #a^m xx a^n = a^(m+n)# works for any integer values of #m# and #n#, positive, negative or #0#.

#color(white)()#
The next level of complexity is:

#(a^m)^n = overbrace(a^m xx a^m xx .. xx a^m)^"n times" = a^(mn)#

For example:

#(2^2)^3 = 4^3 = 64#

#color(white)()#
Finally note that #a^(m^n)# is evaluated from right to left.

That is:

#a^(m^n) = a^((m^n))#

For example:

#2^(2^3) = 2^8 = 256#