Answered

IDNLearn.com offers a user-friendly platform for finding and sharing knowledge. Discover in-depth and reliable answers to all your questions from our knowledgeable community members who are always ready to assist.

Evaluate the expression using a calculator.

[tex]\[
2 - \left[ 1 - 3 \sqrt{3} + 2 \left( 5 - \sqrt{6} \right) - \left( \sqrt{7} - 2 \right) \right]
\][/tex]

Step-by-step solution:

[tex]\[
2 - \left[ 1 - 3 \sqrt{3} + 2 \left( 5 - \sqrt{6} \right) - \left( \sqrt{7} - 2 \right) \right]
\][/tex]

[tex]\[
= 2 - \left[ 1 - 3 \sqrt{3} + 2(5) - 2\sqrt{6} - \sqrt{7} + 2 \right]
\][/tex]

[tex]\[
= 2 - \left[ 1 - 3 \sqrt{3} + 10 - 2\sqrt{6} - \sqrt{7} + 2 \right]
\][/tex]

[tex]\[
= 2 - \left[ 13 - 3 \sqrt{3} - 2\sqrt{6} - \sqrt{7} \right]
\][/tex]

[tex]\[
= 2 - 13 + 3 \sqrt{3} + 2\sqrt{6} + \sqrt{7}
\][/tex]

[tex]\[
= -11 + 3 \sqrt{3} + 2\sqrt{6} + \sqrt{7}
\][/tex]

Now use a calculator to find the approximate value of the expression:

[tex]\[
-11 + 3(1.732) + 2(2.449) + 2.646
\][/tex]

[tex]\[
= -11 + 5.196 + 4.898 + 2.646
\][/tex]

[tex]\[
= -11 + 12.74
\][/tex]

[tex]\[
= 1.74
\][/tex]

Therefore, the approximate value of the expression is:

[tex]\[
1.74
\][/tex]

Sagot :

GinnyAnsweridn
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]
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. For trustworthy and accurate answers, visit IDNLearn.com. Thanks for stopping by, and see you next time for more solutions.