Question

Wat is the simplified expression for 3^-4 • 2^3 • 3^2 over 2^4 • 3^-3

Answer 1

 (3^-4)(2^3)(3^2)  
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(2^4)(3^-3)
I will keep this as simple as possible (for clarity). Any negative exponents should be switched from top to bottom and the negative sign removed from the exponent. 

(3^3)(2^3)(3^2)  
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(2^4)(3^4)

Add the like terms in the numerator

(3^5)(2^3)  
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(2^4)(3^4)

Since we have powers of 3 and powers of 2 in the numerator and denominator we can add them together (just like when we reduce other fractions)  For example, x^4/x => x^3, or x^1/x^6 => 1/x^5

3/2

Final answer 3/2. 

Answer 2

The answer is:  "$\frac{3}{2}$" . 
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          (or, write as: "1½" ; or,  "1.5").
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Explanation:
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We are asked to simplify the given expression:
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  →  $\frac{3^{-4}×2^{3}×3^{2} }{2^{4}×3^{-3}}$  ;
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Note:  In the "numerator" :
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   →  2³  =  2 × 2 × 2  =  8 .

   →  3²  =  3 × 3  =  9 .
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Note:  In the "denominator" :
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   →  2⁴  =  2 × 2 × 2 × 2 = 16 .
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     So, rewrite our expression; substituting "8" for "(2³)";
and substituting "9" for "(3²)" — [in the numerator] ;
and substituting:  "16" for "(2⁴)" — [in the denominator] ;
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   → AS FOLLOWS:
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   →  $\frac{3^{-4}×2^{3}×3^{2} }{2^{4}×3^{-3}}$  ;

          =  $\frac{3^{-4}×8×(9}{16×3^{-3}}$ ;
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   →  Since we have an "8" in the "numerator"; and a "16" in the "denominator" —respectively;  and since both values, taken individually in the numerator—and taken individually in the denominator— are multiplied by other values as isolated numbers;  we can "cancel out" the "8" in the "numerator" to a "1"; and change the "16" in the "denominator" to a "2" ;  since:
            "16÷8 = 2" ; and since "8÷8=1" ;  that is: "8/16 = 1/2".  We can then "eliminate" the "1" in the "numerator";  since in the numerator, there are other values that are multiplied by this "1" ;  & any value multiplied by "1" is equal to that same value.
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So we can rewrite the expression, as follows:
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   →  $\frac{3^{-4}×(9)}{2×3^{-3}}$ ;  

↔ Rearrange and rewrite as follows:
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     →   $\frac{3^{-4}×(9)}{2×3^{-3}}$ 

    =  $\frac{(9) *{3^{-4}}{2×3^{-3}}$ ;
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   → Note the following properties of exponents:
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         ⇒  ($\frac{a} {b}$ ⁿ  = $\frac{ a^{n}}{b^{n}}$   ;  
                      → (b ≠ 0) ; 
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         ⇒  ($a^{m}$ )ⁿ =  a$a^{(m*n)}}$;
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         ⇒  $a^{m} a^{n} = a^{(m+n)}$;

and especially:

         ⇒$\frac{ a^{m}}{ a^{n}} = a^{(m-n)} ; (a \neq 0) ;$;

and especially:

         ⇒  $a^{-n} = \frac{1}{(a^{n) }}$ ;  (a $\neq 0);$);                       If "n" is a positive integer; and if "a" is a non-zero real number. 
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         →  So;  (3⁻⁴) / (3⁻³)  = 3⁽⁽⁻⁴ ⁻ ⁽⁻³⁾⁾ = 3⁽⁻⁴ ⁺ ³⁾ = 3⁻¹  
                                         = $\frac{1}{(3^{1})} = \frac{1}{3} ;$ ;  
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         →  Rewrite the expression:
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         →  $\frac{(9) *{3^{-4}}{2×3^{-3}}$ ; 
 
               =  $\frac{(9*1)}{(2*3)} ; = \frac{9}{6} ; = \frac{(9/3) }{(6/3)} ; = \frac{3}{2}$ ; or; write as: " 1 ½ " ; or, write as: " 1.5 ".
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