Question

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

Answer 1

The answer is: "
`(3)/(2)`" .
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(or, write as: "1½" ; or, "1.5").
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Step-by-step explanation:
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We are asked to simplify the given expression:
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→
`(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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→
`(3^(-4)×2^(3)×3^(2) )/(2^(4)×3^(-3))` ;

=
`(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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→
`(3^(-4)×(9))/(2×3^(-3))` ;

↔ Rearrange and rewrite as follows:
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→
`(3^(-4)×(9))/(2×3^(-3))`

=
;
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→ Note the following properties of exponents:
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⇒ (
`\frac{a} {b}` ⁿ =
`( 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:

⇒
`( a^(m))/( a^(n)) = a^((m-n)) ; (a \\eq 0) ;`;

and especially:

⇒
`a^(-n) = (1)/((a^(n) ))` ; (a
`\\eq 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
=
`(1)/((3^(1))) = (1)/(3) ;` ;
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→ Rewrite the expression:
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→
;

=
`image` ; or; write as: " 1 ½ " ; or, write as: " 1.5 ".
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Answer 2

(3^-4)(2^3)(3^2)
-----------------
(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)
-----------------
(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.