To solve the expression \((8/9) \times (-3/4) \div (1/9) \times (-4/5) \div (3/5)\), we can follow the rules for multiplication and division. First, let's convert the division to multiplication by using the reciprocal. The division \( a \div b \) can be rewritten as \( a \times (1/b) \).
Let's rewrite the expression step-by-step:
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Rewrite division as multiplication by the reciprocal: \[ (8/9) \times (-3/4) \times (9/1) \times (-4/5) \times (5/3) \]
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Now let's organize the expression: \[ = (8/9) \times (-3/4) \times (9/1) \times (-4/5) \times (5/3) \]
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Combine the fractions together: \[ = \frac{8 \times (-3) \times 9 \times (-4) \times 5}{9 \times 4 \times 1 \times 5 \times 3} \]
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Now let's calculate the numerator and the denominator separately.
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Numerator: \[ 8 \times (-3) \times 9 \times (-4) \times 5 = 8 \times 3 \times 9 \times 4 \times 5 \] (Since we are multiplying two negative numbers, the product will be positive).
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Performing the multiplication in parts: \[ = 8 \times 3 = 24 \] \[ = 24 \times 9 = 216 \] \[ = 216 \times 4 = 864 \] \[ = 864 \times 5 = 4320 \]
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Denominator: \[ 9 \times 4 \times 1 \times 5 \times 3 = 9 \times 4 = 36 \] \[ = 36 \times 1 = 36 \] \[ = 36 \times 5 = 180 \] \[ = 180 \times 3 = 540 \]
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Now we combine the results: \[ = \frac{4320}{540} \]
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To simplify \(\frac{4320}{540}\), we can divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 4320 and 540 is 180.
- Divide both by 180: \[ \frac{4320 \div 180}{540 \div 180} = \frac{24}{3} = 8 \]
Thus, the final result of the expression is: \[ \boxed{8} \]