How do exponents raised to another exponent work?

Question

Genesis Amaya · Accepted Answer

See explanation...

Basic positive integer exponents can be seen as repeated multiplication in one way: #a^n = overbrace(a xx a xx .. xx a)^"n times"# Next, we discover: #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)# #color(white)()# The next level of complexity is when a value that has been raised to an exponent is raised to another exponent: #(a^b)^c = a^(bc)# For example, #(2^2)^3 = 4^3 = 64# #color(white)()# The next level is where a value is raised to a value that has been raised to an exponent: Note that #a^(b^c)# is evaluated from right to left, not left to right. So: #a^(b^c) = a^((b^c))# For example, #2^(2^3) = 2^8 = 256#

Roman Alvarez · Answer

undefined Exponents raised to another exponent follow the rule known as the power of a power rule, which states that when you have an exponent raised to another exponent, you multiply the exponents together. In mathematical terms, it can be represented as $ (a^m)^n = a^{m \times n} $. This means that you first raise the base to the power of the inner exponent, and then raise the result to the outer exponent. For example, $ (2^3)^2 $ would be calculated as $ 2^{3 \times 2} = 2^6 $, which equals 64.