Okay, let's break down the given expression step by step to reach the correct result.
### Step 1: Simplify the Inner Expression
We'll start by simplifying the inner expression inside the brackets:
[tex]\[ 2 - [1 - 3\sqrt{3} + 2(5 - \sqrt{6}) - (\sqrt{7} - 2)] \][/tex]
Let's break this down into smaller parts to make the calculations easier.
#### Part 1.1: Simplify Inside the Parentheses
First, consider the innermost parentheses:
[tex]\[ 2(5 - \sqrt{6}) \][/tex]
Now calculate the two terms separately.
- [tex]\( 5 \)[/tex]
- [tex]\(\sqrt{6}\)[/tex] (which is approximately 2.449)
Combining these:
[tex]\[ 2 \times 5 = 10 \][/tex]
[tex]\[ 2 \times \sqrt{6} = 2 \times 2.449 \approx 4.898 \][/tex]
So,
[tex]\[ 2(5 - \sqrt{6}) \approx 10 - 4.898 = 5.102 \][/tex]
Next, consider [tex]\( \sqrt{7} - 2 \)[/tex]:
- [tex]\( \sqrt{7} \)[/tex] which is approximately 2.646
So,
[tex]\[ \sqrt{7} - 2 \approx 2.646 - 2 = 0.646 \][/tex]
#### Part 1.2: Combine the Results
Now substitute these values back into the expression:
[tex]\[ 2 - [1 - 3\sqrt{3} + 5.102 - 0.646] \][/tex]
Let’s simplify the terms inside the brackets:
1. Calculate [tex]\( 3\sqrt{3} \)[/tex]:
- [tex]\( \sqrt{3} \)[/tex] which is approximately 1.732
- So, [tex]\( 3 \times 1.732 \approx 5.196 \)[/tex]
2. Combine all the terms inside the brackets:
[tex]\[ 1 - 5.196 + 5.102 - 0.646 \approx 1 - 5.196 + 5.102 - 0.646 = 0.260 \][/tex]
Thus, the revised expression is:
[tex]\[ 2 - [0.260] \][/tex]
### Step 2: Evaluate the Outer Subtraction
Subtract the result inside the brackets from 2:
[tex]\[ 2 - 0.260 \approx 1.740 \][/tex]
So, the first part results in:
[tex]\[ 1.740 \][/tex]
### Step 3: Simplifying the Second Expression
Next, we handle the following simplified expression:
[tex]\[ 2 - \sqrt{2} - 3\sqrt{3} + 2.55 - 0.65 \][/tex]
1. Calculate each term:
- [tex]\( \sqrt{2} \approx 1.414 \)[/tex]
- [tex]\( 3\sqrt{3} \approx 5.196 \)[/tex] (as previously calculated)
Now substitute these values and compute step by step:
[tex]\[ 2 - 1.414 - 5.196 + 2.55 - 0.65 \][/tex]
### Step 4: Combine All Terms
Let's calculate the value step by step:
[tex]\[ 2 - 1.414 \approx 0.586 \][/tex]
[tex]\[ 0.586 - 5.196 \approx -4.610 \][/tex]
[tex]\[ -4.610 + 2.55 \approx -2.060 \][/tex]
[tex]\[ -2.060 - 0.65 = -2.710 \][/tex]
So, the second part results in:
[tex]\[ -2.710 \][/tex]
### Step 5: Combining the Final Step
Finally, we combine the last simplified expression:
[tex]\[ 2 + 7.74 + 2.55 - 0.65 = 7.64 \][/tex]
So, the final simplified expression results in:
[tex]\[ 7.64 \][/tex]
Therefore, the given expression simplifies to approximately:
[tex]\[ 7.64 \][/tex]
To summarize the step-by-step breakdown results:
- First expression evaluates to approximately: [tex]\( 1.740 \)[/tex]
- Second expression evaluates to approximately: [tex]\( -2.710 \)[/tex]
- Third expression confirms: [tex]\( 7.64 \)[/tex]
