How do you determine which is greater #4^2*4^3 (<=>) 4^5#?

One of the properties of exponents says that
#color(blue)(a^m * a^n = a^(m+n)#

Hence #4^2 * 4^3 = 4^(2+3) = 4^5#

Hence we can say that #color(green)(4^2*4^3 = 4^5#

The rule says that the multiplication of powers of a same base is the base elevated to the sum of the powers. In formulas: #a^b * a^c = a^{b+c}#. The reason is very simple: let's prove that #4^2*4^3=4^5#.

Apply the definition of power: #4^2=4*4#, and #4^3=4*4*4#.

Multiply the two numbers:

#(4*4)*(4*4*4) = 4*4*4*4*4=4^5#

So, no one of the two numbers is greater than the other, they are exactly the same.

Assumption: by the dot you mean multiply.

Consider # 2 times 2#. This may be written as #2^2# ..(1)

Now consider #2 times 2 times 2 times 2#. This may be written as #2^4# ..(2)

Now consider both parts (1) and (2) together:

(2) may be rewritten as #2^2 times 2^2#
but we know that (2) is also #2^4# so:

#2^2 times 2^2 = 2^4#

From this we can deduce that #2^2 times 2^2 = 2^(2+2)#

Ok! now look at the question given:

We have #4^2 times 4^3#

By using the same approach as before this would give us:

#4^(2+3) = 4^5#

In conclusion: both sides of #(<=>)# have the same value so neither is greater.