The answer is: "[tex] \frac{3}{2} [/tex]" .
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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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→ [tex] \frac{3^{-4}×2^{3}×3^{2} }{2^{4}×3^{-3}}
[/tex] ;
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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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→ [tex] \frac{3^{-4}×2^{3}×3^{2} }{2^{4}×3^{-3}} [/tex] ;
= [tex] \frac{3^{-4}×8×(9}{16×3^{-3}} [/tex] ;
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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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→ [tex] \frac{3^{-4}×(9)}{2×3^{-3}} [/tex] ;
↔ Rearrange and rewrite as follows:
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→ [tex] \frac{3^{-4}×(9)}{2×3^{-3}} [/tex]
= [tex] \frac{(9) *{3^{-4}}{2×3^{-3}} [/tex] ;
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→ Note the following properties of exponents:
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⇒ ([tex] \frac{a} {b} [/tex] ⁿ = [tex] \frac{ a^{n}}{b^{n}} [/tex] ;
→ (b ≠ 0) ;
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⇒ ([tex] a^{m} [/tex] )ⁿ = a[tex] a^{(m*n)}} [/tex];
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⇒ [tex] a^{m} a^{n} = a^{(m+n)} [/tex];
and especially:
⇒[tex] \frac{ a^{m}}{ a^{n}} = a^{(m-n)} ; (a \neq 0) ;[/tex];
and especially:
⇒ [tex] a^{-n} = \frac{1}{(a^{n) }} [/tex] ; (a [tex] \neq 0); [/tex]); 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
= [tex] \frac{1}{(3^{1})} = \frac{1}{3} ;[/tex] ;
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→ Rewrite the expression:
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→ [tex] \frac{(9) *{3^{-4}}{2×3^{-3}} [/tex] ;
= [tex] \frac{(9*1)}{(2*3)} ;
= \frac{9}{6} ;
= \frac{(9/3) }{(6/3)} ;
= \frac{3}{2}[/tex] ; or; write as: " 1 ½ " ; or, write as: " 1.5 ".
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