To solve the expression \((-2) \times (15) \div (23) \times (-56) \div (34)\), we will follow the order of operations and simplify step-by-step.
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Calculate the first part: \((-2) \times (15)\) \[ -2 \times 15 = -30 \]
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Now, divide by \(23\): \[ -30 \div 23 = -\frac{30}{23} \]
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Next, multiply by \((-56)\): \[ -\frac{30}{23} \times (-56) = \frac{30 \times 56}{23} \]
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Calculate \(30 \times 56\): \[ 30 \times 56 = 1680 \] Thus, we have: \[ \frac{1680}{23} \]
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Finally, divide by \(34\): \[ \frac{1680}{23} \div 34 = \frac{1680}{23} \times \frac{1}{34} = \frac{1680}{23 \times 34} \]
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Calculate \(23 \times 34\): \[ 23 \times 34 = 782 \] Therefore: \[ \frac{1680}{782} \]
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Now we simplify \(\frac{1680}{782}\). We can find the GCD (Greatest Common Divisor) of 1680 and 782 to reduce it:
- The factors of 1680 are: \(2^4 \times 3 \times 5 \times 7\)
- The factors of 782 are: \(2 \times 391\) (where 391 is a prime number)
The GCD is \(2\). Dividing both the numerator and denominator by \(2\): \[ \frac{1680 \div 2}{782 \div 2} = \frac{840}{391} \]
This doesn't exactly match any of the provided options, but we can calculate the decimal or mixed number form to compare.
Calculating the decimal for \(\frac{840}{391}\): \[ 840 \div 391 \approx 2.148 \]
Expressed as a mixed number, \(2\) remainder \(58\) gives: \[ 2 + \frac{58}{391} \]
Let’s check other options to find a close match:
- \(16\)
- \(-23\)
- \(-16\)
- \(\frac{2}{3}\)
- \(\frac{23}{3}\)
None of these are direct alternatives of our finding, but since all steps are correct; if amendments or different approaches lead to one of these causes, please clarify. However, based on exact calculations, we conclude with the right process yielding thus \( \frac{840}{391} \).