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QuestionCalculate the value of □⋅124⋅5−3⋅928⋅36⋅135−1\square \cdot \frac{12^{4} \cdot 5^{-3} \cdot 9^{2}}{8 \cdot 3^{6} \cdot 135^{-1}}.

Studdy Solution

STEP 1

Assumptions1. We need to simplify the given expression. . All the operations are multiplication and division, which are commutative and associative. This means we can rearrange and regroup the terms as needed.

STEP 2

First, let's rewrite the expression in a more manageable form. We can do this by separating the terms with the same base.
124⋅5−⋅928⋅6⋅135−1=28⋅4⋅5−⋅4⋅2−62⋅6⋅−1⋅5−1⋅2−\frac{12^{4} \cdot5^{-} \cdot9^{2}}{8 \cdot^{6} \cdot135^{-1}} = \frac{2^{8} \cdot^{4} \cdot5^{-} \cdot^{4} \cdot2^{-6} }{2^{} \cdot^{6} \cdot^{-1} \cdot5^{-1} \cdot2^{-}}

STEP 3

Now, let's simplify the expression by combining the terms with the same base.
28⋅3⋅5−3⋅3⋅2−623⋅36⋅3−1⋅5−1⋅2−3=28−3−6⋅3+−6+1⋅5−3+1\frac{2^{8} \cdot3^{} \cdot5^{-3} \cdot3^{} \cdot2^{-6} }{2^{3} \cdot3^{6} \cdot3^{-1} \cdot5^{-1} \cdot2^{-3}} =2^{8-3-6} \cdot3^{+-6+1} \cdot5^{-3+1}

STEP 4

Calculate the exponents for each base.
28−3−6⋅34+4−6+1⋅−3+1=2−1⋅33⋅−22^{8-3-6} \cdot3^{4+4-6+1} \cdot^{-3+1} =2^{-1} \cdot3^{3} \cdot^{-2}

STEP 5

Convert the negative exponents to positive by taking the reciprocal.
2−1⋅33⋅5−2=12⋅33⋅1522^{-1} \cdot3^{3} \cdot5^{-2} = \frac{1}{2} \cdot3^{3} \cdot \frac{1}{5^{2}}

STEP 6

Calculate the value of each term.
12⋅33⋅152=12⋅27⋅125\frac{1}{2} \cdot3^{3} \cdot \frac{1}{5^{2}} = \frac{1}{2} \cdot27 \cdot \frac{1}{25}

STEP 7

implify the expression by multiplying the terms.
12⋅27⋅125=2750\frac{1}{2} \cdot27 \cdot \frac{1}{25} = \frac{27}{50}So, the simplified form of the given expression is 2750\frac{27}{50}.

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