How do you simplify #(5^8)^3#?

Exponent rules or sometimes called as laws of exponents.
You can go over the rules and would be in a position to solve many such problems as you have shared.

Let us understand exponents in easier manner.

#a^1=a#
#a^2=a*a#
#a^3=a*a*a#

You can see the exponent denotes the number of time the base is to be multiplied with itself.

A very common rule is

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

You should be familiar with it, if not take some time and go over the rules it would be such a life saver later on in Maths.

Now let us come problem similar to ours.

Example : #(a^2)^3#

This we can write as #a^2*a^2*a^2# that is what we understand by exponents.

Now applying the rule we can see

#(a^2)^3 = a^2*a^2*a^2#
#(a^2)^3 = a^(2+2+2)#
#(a^2)^3 = a^6#

Now I would like to point out that the product of the two exponents here that is #2# and #3# also gives us #6#

To generalize

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

The above rule can be used for our problem.

#(5^8)^3#

#=5^(8xx3)#

#=5^24# Answer.

We understand that raising something to a power produces a product of that number and itself as many times as the number the power is.
e.g. #6^3=6*6*6#
So #(5^8)^3=5^8*5^8*5^8#

By our law of indicies: #x^n*x^m=x^(n+m)#

We therefore have: #(5^8)^3=5^8*5^8*5^8=5^(8+8+8)=5^24#

Taking it even simpler, #5^8=5*5*5*5*5*5*5*5#

So: #5^8*5^8*5^8=5*5*5*5*5*5*5*5*5*5*5*5*5*5*5*5*5*5*5*5*5*5*5*5=5^24#

Overall, the rule is, if you have a number raised to a power, and both the number and power are raised to another power, you can simply multiply the two powers together to get the new power for the number.

e.g: #(x^m)^n=x^(mn)#