Question

How do you simplify `((3^6)^n times (81)^(2n)) / (3^n)^4`?

Answer 1

`= color(green)(3^(10n)`

Explanation:

`((3^6)^n xx (81)^(2n)) /((3^n)^4`

• Simplifying `81` by prime factorisation (expressing a number as a product of its prime factors):

`81 = 3 * 3 * 3 * 3 = color(blue)(3^4`

So, `81 ^(2n) = (3^4 )^(2n)`

• Applying below mentioned property to the expression:
  `color(blue)(a^m)^n =a ^(mn)`

• `(3^4 )^(2n) = color(green)( 3 ^(8n)`

• `(3^6)^n = 3^(6n)`

• `(3^n)^4 = 3^(4n)`

The expression can now be written as:

`((3^6)^n xx (81)^(2n)) /((3^n)^4 ) = (3^(6n) xx color(green)( 3 ^(8n))) /(3^ ( 4n)`

• Applying below mentioned property to the numerator:
  `color(blue)(a^m xx a^n = a ^(m+n)`

`=3^((6n + 8n)) /(3^ ( 4n)`

`=3^(14n) /(3^ ( 4n)`

• Applying below mentioned property to the expression:
  `color(blue)(a^m / a ^n = a ^(m-n)`

`=3^((14n - 4n))`

`= color(green)(3^(10n)`