15  mod  4    -   37  +  42   /  (15  -  4) * 4  ^ 3   /  (15 + 42  mod 27)

Let's break down the given expression step by step:

\[
15 \mod 4 - 37 + \frac{42}{(15 - 4)} \times 4^3 \div (15 + 42 \mod 27)
\]

### Step 1: Resolve the modulus operations
1. \( 15 \mod 4 = 15 - 4 \times \left\lfloor \frac{15}{4} \right\rfloor = 15 - 4 \times 3 = 15 - 12 = 3 \)
2. \( 42 \mod 27 = 42 - 27 \times \left\lfloor \frac{42}{27} \right\rfloor = 42 - 27 = 15 \)

So the expression becomes:
\[
3 - 37 + \frac{42}{(15 - 4)} \times 4^3 \div (15 + 15)
\]

### Step 2: Simplify the subtraction and addition inside the parentheses
1. \( 15 - 4 = 11 \)
2. \( 15 + 15 = 30 \)

So now the expression is:
\[
3 - 37 + \frac{42}{11} \times 4^3 \div 30
\]

### Step 3: Evaluate the exponent
\[
4^3 = 4 \times 4 \times 4 = 64
\]

Now the expression is:
\[
3 - 37 + \frac{42}{11} \times 64 \div 30
\]

### Step 4: Perform division and multiplication
1. \( \frac{42}{11} \approx 3.8182 \)
2. \( 3.8182 \times 64 = 244.3648 \)
3. \( \frac{244.3648}{30} \approx 8.1455 \)

Now the expression becomes:
\[
3 - 37 + 8.1455
\]

### Step 5: Simplify the remaining terms
\[
3 - 37 + 8.1455 = -34 + 8.1455 = -25.8545
\]

### Final Answer:
\[
\boxed{-25.8545}
\]

Would you like more details on any of these steps, or have any further questions?

Here are 5 related questions:
1. How do you calculate modulus in different bases?
2. How does order of operations affect the outcome of complex expressions?
3. How can you simplify expressions involving both division and exponents?
4. What are common pitfalls when using mixed operations (addition, multiplication, etc.)?
5. How do modulus and division interact in mathematical expressions?

**Tip:** Always follow the order of operations (PEMDAS/BODMAS) to avoid mistakes in complex calculations.

15 mod 4 - 37 + 42 / (15 - 4) * 4 ^ 3 / (15 + 42 mod 27)

Explore the detailed solution for the expression 15 mod 4 - 37 + 42 / (15 - 4) * 4 ^ 3 / (15 + 42 mod 27). This problem incorporates modulus, exponentiation, division, and order of operations. Learn step-by-step how to simplify the expression, suitable for high school students.

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