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Use the properties of operations and rules for multiplying and dividing signed numbers to solve:

(−4)×(75)×(−34)÷(715)

(1 point)
Responses

−14775
negative Start Fraction 147 over 75 End Fraction

14775
Start Fraction 147 over 75 End Fraction

−9
negative 9

9

1 answer

To solve the expression \( (-4) \times (75) \times (-34) \div (715) \), we will follow the order of operations and the signs for multiplication and division.

  1. Multiply the first two numbers: \[ (-4) \times (75) = -300 \]

  2. Multiply the result by \(-34\): \[ -300 \times (-34) = 10200 \] (The product of two negative numbers is positive.)

  3. Divide the result by \(715\): \[ \frac{10200}{715} \]

Now, we need to simplify \( \frac{10200}{715} \). We can factor both numbers to simplify the fraction. To do this, let's find the greatest common divisor (GCD) of 10200 and 715:

  • The prime factorization of \(10200\):

    • \(10200 = 102 \times 100 = 102 \times 10^2 = 102 \times (2 \times 5)^2\)
    • The factors can also be resolved into simpler forms if desired.

  • The prime factorization of \(715\):

    • \(715 = 5 \times 143 = 5 \times (11 \times 13)\)

Now, we know that both 10200 and 715 share a factor of 5.

Dividing both by 5:

  • \(10200 \div 5 = 2040\)
  • \(715 \div 5 = 143\)

Thus: \[ \frac{10200}{715} = \frac{2040}{143} \]

Since 2040 and 143 share no more common factors, this fraction is in its simplest form.

Therefore, the final answer is: \[ \frac{2040}{143} \] Converting this to a decimal or mixed number would be unnecessary, as the current response options do not reflect that. However, since the simplified form is not among the responses, we note that the values directly provided were incorrect for consideration.

Out of the provided options:

  • The answer is not on the list, and the closest comprehension would be to simply note the frustration in observing simpler fraction reduction wasn't readily available.

If you have to choose from the responses, the closest numerical approximation would suggest that \( \frac{2040}{143} \) leads towards a positive and larger approximation towards 9, but explicit contextual mapping to responses was flawed in the transcription. Hence, leading notes towards the actual simplification as the most efficient takeaway for proper verification and alignment when provided choices do not match correct computed outputs readily.