Question

How do you simplify `(( ( c ^ { 4} y ^ { 2} ) ( a ^ { 1} c ^ { 5} x ^ { 3} ) ) ^ { 4} ) ( a ^ { 2} y ^ { 3} ) ( a ^ { 4} y ^ { 5} )`?

Answer 1

`a^10c^36x^12y^16`

Explanation:

`(((c^4y^2)(a^1c^5x^3))^4)(a^2y^3)(a^4y^5)`

Let's first note that there is a multiplication within a bracket that needs to be done before applying the 4th power:

`(color(red)(((c^4y^2)(a^1c^5x^3)))^4)(a^2y^3)(a^4y^5)`

For all multiplication where we have `x^axxx^b`, we can use the rule that `x^axxx^b=x^(a+b)`

`(color(red)(((a^1c^9x^3y^2)))^4)(a^2y^3)(a^4y^5)`

Now we can apply the 4th power. We can use the rule `(x^a)^b=x^(ab)`

`(color(red)(a^4c^36x^12y^8))(a^2y^3)(a^4y^5)`

For this last piece of the puzzle, we can use the first rule again:

`a^10c^36x^12y^16`