Simplify the expression. (−5a5b5)2(a3b3)5 25a15b15 25a25b25 −25a25b25

The correct answer is:  [B]:  " 25 a²⁵ b²⁵ " .

Explanation:

Given the expression: 

       →  " (−5a⁵b⁵)² (a³b³)⁵  " ;   Simplify.

Let us being by examining:

       →    "(−5a⁵b⁵)² " . 

→  "(−5a⁵b⁵)²  = (-5)² * (a⁵)² * (b⁵)²  = (-5)(-5) * a⁽⁵ˣ²⁾ * b⁽⁵ˣ²⁾  = 25a⁽¹⁰⁾b⁽¹⁰⁾ ;

{Note the following properties of exponents:
    (xy)ⁿ = xⁿ * yⁿ ; 

    (xᵃ)ᵇ = x⁽ᵃ * ᵇ) ; 

    (xᵃ) * (xᵇ) = x⁽ᵃ ⁺ ᵇ⁾ .}.


Then, we examine:

      →    "(a³b³)⁵ " .

→  "(a³b³)⁵ = a⁽³ˣ⁵⁾b⁽³ˣ⁵⁾ = a⁽¹⁵⁾b⁽¹⁵⁾ .


So:   " (−5a⁵b⁵)² (a³b³)⁵ = (-5)a⁽¹⁰⁾b⁽¹⁰⁾ * a⁽¹⁵⁾b⁽¹⁵⁾  " ; 

Now, we simplify:

          →  " 25a⁽¹⁰⁾b⁽¹⁰⁾ * a⁽¹⁵⁾b⁽¹⁵⁾ " ; 

→  " 25a⁽¹⁰⁾b⁽¹⁰⁾ * a⁽¹⁵⁾b⁽¹⁵⁾ ;
 
               =  25a⁽¹⁰⁾ a⁽¹⁵⁾b⁽¹⁰⁾ b⁽¹⁵⁾  ;

               =  25a⁽¹⁰ ⁺¹⁵⁾ b⁽¹⁰⁺¹⁵⁾ ;

               =  25a⁽²⁵⁾ b⁽²⁵⁾ ; 

  →  which is:  Answer choice:  [B]:  " 25 a²⁵ b²⁵ " .