Question

How do you simplify `(7^4)^3`?

Answer 1

`7^(12)`

Explanation:

Using the `color(blue)"law of exponents"`

`color(orange)"Reminder " color(red)(bar(ul(|color(white)(2/2)color(black)(a^mxxa^n=a^(m+n))color(white)(2/2)|)))`
which can be extended to the product of more than 2 terms.

`rArr(7^4)^3=7^4xx7^4xx7^4`

`=7^(4+4+4)=7^(12)`

Note that the exponent 12, is also obtained by `color(blue)"multiplying"` 3 and 4 together. This leads us to a further law of exponents.

`"That is " color(red)(bar(ul(|color(white)(2/2)color(black)((a^m)^n=a^(mn))color(white)(2/2)|)))`

`rArr(7^4)^3=7^(4xx3)=7^(12)`

Answer 2

`(7^4)^3 = 7^12`

Explanation:

Note that if `a` is any number and `m, n` are positive integers then:

`(a^m)^n=overbrace(a^mxxa^mxx...xxa^m)^"n times"`

`color(white)((a^m)^n)=overbrace(overbrace(axxaxx...xxa)^"m times"xxoverbrace(axxaxx...xxa)^"m times"xx...xxoverbrace(axxaxx...xxa)^"m times")^"n times"`

`color(white)((a^m)^n)=overbrace(axxaxx...xxa)^"mn times"`

`color(white)((a^m)^n)=a^(mn)`

So in our example:

`(7^4)^3 = 7^(4*3) = 7^12`