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)`