Question

What is 7 over (fraction bar) (8×4)+90÷9 ÷ (6^3)^-2 × 9 over (fraction bar) 6(5+4) = ?????

Answer 1

The answer is:  " 36 " .
_______________________________
Note:  " (8 * 4) + 90 ÷ 9 " =  " (8 * 4) + (90 ÷ 9) " = "32 + 10 = 42" .

We can rewrite the "fraction":
 
7 / [ (8 * 4) + 90 ÷ 9 ] ;  as:  $\frac{7}{42}$ ;  

and can reduce this value to (and rewrite as):  " $\frac{1}{6}$ " ; 
    { since:  " $\frac{7}{42}$  =  $\frac{7/7}{42/7}$  = $\frac{1}{6}$" .}.

Note:  The "fraction" : $\frac{9}{6(5+4)}$ ;  the "9/6" cancel to "3/2" ; 
  {Since:  "9÷3 = 3" ; and since:  "6÷3 = 2" ).

So rewrite:  $\frac{9}{6(5+4)}$ ; as:  

$\frac{3}{2(5+4)}$ ;

Note:  $\frac{3}{2(5+4)}$  ;

       =   $\frac{3}{2(9)}$ ;

       =   $\frac{3}{18}$ ;

→  Note:   $\frac{3}{18}$ ;  can be reduced to; & rewritten as:

             " $\frac{1}{6}$ " ;

→ {since: " $\frac{3}{18}$ =  $\frac{3/3}{18/3}$ ;
               
                                                      =   $\frac{1}{6}$ ;
______________________________

Note that: (6²) ⁻² = 6⁽ ² * ⁻²⁾ = 6⁻ ⁴  ;
    
                            =  $\frac{1}{6^4}$ ;
_________________________________________________________
So, we can rewrite our expression as:

 $\frac{1}{6}$  ÷  $\frac{1}{6^4}$  *  $\frac{1}{6}$ ;
_________________________________________________________
Note:  
_________________________________________________________
 $\frac{1}{6}$  ÷  $\frac{1}{6^4}$  *  $\frac{1}{6}$ ;

= ( $\frac{1}{6}$  ÷  $\frac{1}{6^4}$ ) *  $\frac{1}{6}$ ; 

= ( $\frac{1}{6}$  *   $\frac{6^4}{1}$ ) *  $\frac{1}{6}$ ;

________________________________________________________
Start with :
________________________________________________________
   → ( $\frac{1}{6}$  *   $\frac{6^4}{1}$ ) ;

            The "6" in  the  " $\frac{1}{6}$ " ;  cancels to a "1" ; 
       and the "6⁴ " in the "  $\frac{6^4}{1}$ ) " ; cancels to a "6³ " ;

         → {Since:  "6⁴ / 6  =  6⁴ / 6¹  =  6⁽⁴ ⁻ ¹⁾  = 6³ ;

         → and we have:   $\frac{1}{1}$  *  $\frac{6^3}{1}$ ;
 
         →  The " $\frac{1}{1}$ " can be eliminated; since "1÷1= 1 " ;
               and since: any value, divided by "1", equal that same value; & since "any                       value, multiplied by "1"; equals that same value" ;
________________________________________________________
 We have:  " 6³ " .
________________________________________________________

Now, we multiplied this value, " 6³ " , by  " $\frac{1}{6}$ " ;

→  6³  *  $\frac{1}{6}$  ;

     =  $\frac{6^3}{1}$ * $\frac{1}{6}$  ;

Note:  The "6³ " cancels out to "6² " ;  and the "6" cancels out to "1" ; 

          {since:  "6³ ÷ 6 = 6³ ÷ 6¹ = 6⁽³ ⁻ ¹⁾ = 6² " ;
            and since:  "6 ÷ 6 = 1 "  ; ________________________________________________________
→ The answer is:  6² = 6 * 6 = 36 ;
________________________________________________________
The answer is:  " 36 " .
________________________________________________________