2(-3)(-1)^3-7(-2)/6(-1)^4+8(-2), 5(-1)-2(6(-3)-(-2)^3), 12(-3)-4(-2)^2+7(-1)^6/-3^2

Let's break down and solve each of the expressions step by step.

### Expression 1: 
\[ 2(-3)(-1)^3 - 7(-2)/6(-1)^4 + 8(-2) \]

1. Calculate \((-1)^3 = -1\).
2. Calculate \(2(-3)(-1) = 6\).
3. Calculate \((-1)^4 = 1\).
4. Calculate \(7(-2)/6(1) = -14/6 = -7/3\).
5. Calculate \(8(-2) = -16\).

Now, substitute these into the expression:
\[
6 - \frac{7}{3} - 16 = 6 - \frac{7}{3} - 16 = \frac{18}{3} - \frac{7}{3} - \frac{48}{3} = \frac{18 - 7 - 48}{3} = \frac{-37}{3} = -\frac{37}{3}
\]

### Expression 2: 
\[ 5(-1) - 2(6(-3) - (-2)^3) \]

1. Calculate \(5(-1) = -5\).
2. Calculate \(6(-3) = -18\).
3. Calculate \((-2)^3 = -8\).
4. Calculate \(-18 - (-8) = -18 + 8 = -10\).
5. Calculate \( -2(-10) = 20\).

Now, substitute these into the expression:
\[
-5 + 20 = 15
\]

### Expression 3: 
\[ 12(-3) - 4(-2)^2 + \frac{7(-1)^6}{-3^2} \]

1. Calculate \(12(-3) = -36\).
2. Calculate \((-2)^2 = 4\) and then \(4(-4) = -16\).
3. Calculate \((-1)^6 = 1\).
4. Calculate \(-3^2 = 9\) and then \(\frac{7(1)}{9} = \frac{7}{9}\).

Now, substitute these into the expression:
\[
-36 - 16 + \frac{7}{9} = -52 + \frac{7}{9} = -\frac{468}{9} + \frac{7}{9} = -\frac{461}{9}
\]

### Final Answers:

1. Expression 1: \(-\frac{37}{3}\)
2. Expression 2: \(15\)
3. Expression 3: \(-\frac{461}{9}\)

Do you want details or have any questions?

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Here are 5 related questions:

1. How do you simplify expressions involving exponents with negative bases?
2. What is the rule for multiplying negative numbers?
3. How do you handle order of operations when both division and multiplication are involved?
4. How can fractional results be simplified further in expressions?
5. What is the significance of evaluating expressions with negative bases raised to even and odd powers?

**Tip:** Always pay attention to the signs when dealing with negative numbers in calculations. Misplacing a sign can lead to incorrect results.

Explore solutions to complex arithmetic and exponent expressions with step-by-step explanations. This problem involves arithmetic operations with negative numbers, exponents, and fractions. Suitable for high school students learning advanced arithmetic concepts.

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