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Sagot :
Let's solve the expression [tex]\(2^2(3-8) \div 5 - 1\)[/tex] step by step, identifying the point where Shelly made her mistake.
### Step-by-Step Solution:
1. Step 1: Evaluate the expression inside the parentheses
- The expression inside the parentheses is [tex]\((3 - 8)\)[/tex].
- Calculation: [tex]\(3 - 8 = -5\)[/tex].
- Expression now: [tex]\(2^2(-5) \div 5 - 1\)[/tex].
2. Step 2: Evaluate the exponent
- The exponent is [tex]\(2^2\)[/tex].
- Calculation: [tex]\(2^2 = 4\)[/tex].
- Expression now: [tex]\(4(-5) \div 5 - 1\)[/tex].
3. Step 3: Perform the multiplication
- Multiply [tex]\(4\)[/tex] by [tex]\(-5\)[/tex].
- Calculation: [tex]\(4 \times (-5) = -20\)[/tex].
- Expression now: [tex]\(-20 \div 5 - 1\)[/tex].
4. Step 4: Perform the division
- Divide [tex]\(-20\)[/tex] by [tex]\(5\)[/tex].
- Calculation: [tex]\(-20 \div 5 = -4\)[/tex].
- Expression now: [tex]\(-4 - 1\)[/tex].
5. Step 5: Perform the subtraction
- Subtract [tex]\(1\)[/tex] from [tex]\(-4\)[/tex].
- Calculation: [tex]\(-4 - 1 = -5\)[/tex].
- Final result: [tex]\(-5\)[/tex].
At this point, we have the final result and all intermediate steps:
- Step 1: [tex]\((3 - 8) = -5\)[/tex]
- Step 2: [tex]\(2^2 = 4\)[/tex]
- Step 3: [tex]\(4 \times (-5) = -20\)[/tex]
- Step 4: [tex]\(-20 \div 5 = -4\)[/tex]
- Step 5: [tex]\(-4 - 1 = -5\)[/tex]
### Identifying Shelly's Mistake:
Let's review Shelly's steps:
| Step | Shelly's Work | Mistake? |
|------|---------------|----------|
| 1 | [tex]\(2^2(3-8) \div 5 - 1\)[/tex] | No |
| 2 | [tex]\(2^2(-5) \div 5 - 1\)[/tex] | No |
| 3 | [tex]\(4(-5) \div 5 - 1\)[/tex] | No |
| 4 | [tex]\(-20 \div 4\)[/tex] | Yes, it should be [tex]\(-20 \div 5\)[/tex] |
| 5 | [tex]\(-5\)[/tex] | Incorrect due to error in Step 4 |
Shelly's mistake occurred in Step 4, where she incorrectly performed the division by using [tex]\(4\)[/tex] instead of [tex]\(5\)[/tex]. The correct division should be [tex]\(-20 \div 5\)[/tex], not [tex]\(-20 \div 4\)[/tex]. This mistake led to an incorrect final answer.
### Step-by-Step Solution:
1. Step 1: Evaluate the expression inside the parentheses
- The expression inside the parentheses is [tex]\((3 - 8)\)[/tex].
- Calculation: [tex]\(3 - 8 = -5\)[/tex].
- Expression now: [tex]\(2^2(-5) \div 5 - 1\)[/tex].
2. Step 2: Evaluate the exponent
- The exponent is [tex]\(2^2\)[/tex].
- Calculation: [tex]\(2^2 = 4\)[/tex].
- Expression now: [tex]\(4(-5) \div 5 - 1\)[/tex].
3. Step 3: Perform the multiplication
- Multiply [tex]\(4\)[/tex] by [tex]\(-5\)[/tex].
- Calculation: [tex]\(4 \times (-5) = -20\)[/tex].
- Expression now: [tex]\(-20 \div 5 - 1\)[/tex].
4. Step 4: Perform the division
- Divide [tex]\(-20\)[/tex] by [tex]\(5\)[/tex].
- Calculation: [tex]\(-20 \div 5 = -4\)[/tex].
- Expression now: [tex]\(-4 - 1\)[/tex].
5. Step 5: Perform the subtraction
- Subtract [tex]\(1\)[/tex] from [tex]\(-4\)[/tex].
- Calculation: [tex]\(-4 - 1 = -5\)[/tex].
- Final result: [tex]\(-5\)[/tex].
At this point, we have the final result and all intermediate steps:
- Step 1: [tex]\((3 - 8) = -5\)[/tex]
- Step 2: [tex]\(2^2 = 4\)[/tex]
- Step 3: [tex]\(4 \times (-5) = -20\)[/tex]
- Step 4: [tex]\(-20 \div 5 = -4\)[/tex]
- Step 5: [tex]\(-4 - 1 = -5\)[/tex]
### Identifying Shelly's Mistake:
Let's review Shelly's steps:
| Step | Shelly's Work | Mistake? |
|------|---------------|----------|
| 1 | [tex]\(2^2(3-8) \div 5 - 1\)[/tex] | No |
| 2 | [tex]\(2^2(-5) \div 5 - 1\)[/tex] | No |
| 3 | [tex]\(4(-5) \div 5 - 1\)[/tex] | No |
| 4 | [tex]\(-20 \div 4\)[/tex] | Yes, it should be [tex]\(-20 \div 5\)[/tex] |
| 5 | [tex]\(-5\)[/tex] | Incorrect due to error in Step 4 |
Shelly's mistake occurred in Step 4, where she incorrectly performed the division by using [tex]\(4\)[/tex] instead of [tex]\(5\)[/tex]. The correct division should be [tex]\(-20 \div 5\)[/tex], not [tex]\(-20 \div 4\)[/tex]. This mistake led to an incorrect final answer.
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