What order of operations is necessary in order for "3+4*1000/10-5^3*30-5" to equal 803?

To find the order of operations that makes the expression \(3 + 4 \times 1000 / 10 - 5^3 \times 30 - 5\) equal to 803, let's evaluate the expression step by step following the standard order of operations (PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

### Step 1: Evaluate Exponents
\[
3 + 4 \times 1000 / 10 - 5^3 \times 30 - 5
\]
Here, \(5^3 = 125\). So the expression becomes:
\[
3 + 4 \times 1000 / 10 - 125 \times 30 - 5
\]

### Step 2: Perform Multiplication and Division from left to right
1. Calculate \(4 \times 1000 = 4000\):
\[
3 + 4000 / 10 - 125 \times 30 - 5
\]

2. Next, \(4000 / 10 = 400\):
\[
3 + 400 - 125 \times 30 - 5
\]

3. Then, calculate \(125 \times 30 = 3750\):
\[
3 + 400 - 3750 - 5
\]

### Step 3: Perform Addition and Subtraction from left to right
1. \(3 + 400 = 403\):
\[
403 - 3750 - 5
\]

2. \(403 - 3750 = -3347\):
\[
-3347 - 5
\]

3. Finally, \(-3347 - 5 = -3352\).

The expression evaluates to \(-3352\), which is not equal to 803.

### Modification to achieve 803:
The goal is to manipulate the order of operations (possibly by introducing parentheses) to obtain the result 803. Here's how you could modify the expression:

\[
3 + 4 \times \left(\frac{1000}{10}\right) - \left(5^3 \times 30\right) - 5
\]

Let's compute this adjusted expression:

1. Inside the parentheses, \(1000 / 10 = 100\), so:
\[
3 + 4 \times 100 - \left(5^3 \times 30\right) - 5
\]

2. Now \(4 \times 100 = 400\), so:
\[
3 + 400 - \left(5^3 \times 30\right) - 5
\]

3. Next, \(5^3 = 125\), and \(125 \times 30 = 3750\), so:
\[
3 + 400 - 3750 - 5
\]

4. Perform the addition and subtraction from left to right:
\[
403 - 3750 - 5 = -3347 - 5 = -3352
\]

Since this didn't work, it suggests a different grouping might be necessary. Here's one approach:

\[
\left(3 + 4\right) \times \left(\frac{1000}{10}\right) - 5^3 \times 30 - 5
\]

1. Inside the first parentheses, \(3 + 4 = 7\), so:
\[
7 \times \left(\frac{1000}{10}\right) - 5^3 \times 30 - 5
\]

2. Then, \(1000 / 10 = 100\), so:
\[
7 \times 100 - 5^3 \times 30 - 5
\]

3. Multiply \(7 \times 100 = 700\):
\[
700 - 5^3 \times 30 - 5
\]

4. Then, \(125 \times 30 = 3750\):
\[
700 - 3750 - 5
\]

5. Subtract:
\[
700 - 3750 = -3050 - 5 = -3055
\]

### Conclusion:
The expression \(3 + 4 \times 1000 / 10 - 5^3 \times 30 - 5\) cannot be directly modified to equal 803 without changing the actual numbers or introducing a different operation.

If you have specific operations in mind that should be grouped differently, that could potentially give the result you're looking for. Otherwise, it's necessary to reconsider the original expression or the desired result.

Would you like to explore different modifications, or clarify any other part?

---

**5 Related Questions:**
1. How does the order of operations work with more complex expressions, like those with nested parentheses?
2. What happens if the expression has more than one exponentiation, such as \(a^b^c\)?
3. How does changing a subtraction to an addition in the middle of an expression affect the result?
4. Can you show how adding parentheses can change the outcome of a basic arithmetic expression?
5. What are some common mistakes made when calculating expressions using the order of operations?

**Tip:** Always double-check the result by computing step-by-step to avoid misinterpreting the order of operations.

What order of operations is necessary in order for '3 + 4 * 1000 / 10 - 5^3 * 30 - 5' to equal 803?

Explore the detailed step-by-step process to determine the correct order of operations for the expression '3 + 4 * 1000 / 10 - 5^3 * 30 - 5' to achieve a result of 803. This problem covers key arithmetic concepts including exponents, multiplication, and division, suitable for students in Grades 6-8.

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