Sure, let's simplify each part step by step.
### Part i: Simplify [tex]\(5 \sqrt{2} + 7 \sqrt{2} - 4 \sqrt{2}\)[/tex]
Start by combining the terms with [tex]\(\sqrt{2}\)[/tex]:
[tex]\[
5 \sqrt{2} + 7 \sqrt{2} - 4 \sqrt{2} = (5 + 7 - 4) \sqrt{2} = 8 \sqrt{2}
\][/tex]
The result is:
[tex]\[
8 \sqrt{2} \approx 11.313708498984763
\][/tex]
### Part ii: Simplify [tex]\((4 + 3 \sqrt{3})(4 + 4 \sqrt{3})\)[/tex]
Use the distributive property (also known as the FOIL method for binomials) to expand the expression:
[tex]\[
(4 + 3 \sqrt{3})(4 + 4 \sqrt{3}) = 4 \cdot 4 + 4 \cdot 4 \sqrt{3} + 3 \sqrt{3} \cdot 4 + 3 \sqrt{3} \cdot 4 \sqrt{3}
\][/tex]
Now, calculate each term individually:
[tex]\[
4 \times 4 = 16
\][/tex]
[tex]\[
4 \times 4 \sqrt{3} = 16 \sqrt{3}
\][/tex]
[tex]\[
3 \sqrt{3} \times 4 = 12 \sqrt{3}
\][/tex]
[tex]\[
3 \sqrt{3} \times 4 \sqrt{3} = 12 \times 3 = 36
\][/tex]
Combine the terms:
[tex]\[
16 + 16 \sqrt{3} + 12 \sqrt{3} + 36
\][/tex]
Combine the like terms:
[tex]\[
16 + 36 + (16+12) \sqrt{3} = 52 + 28 \sqrt{3}
\][/tex]
The result is:
[tex]\[
52 + 28 \sqrt{3} \approx 80.08587988004845
\][/tex]
### Part iii: Simplify [tex]\(5 \sqrt{3} - \sqrt{27} - 7 \sqrt{3}\)[/tex]
First, simplify [tex]\(\sqrt{27}\)[/tex]:
[tex]\[
\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3 \sqrt{3}
\][/tex]
Substitute this back into the expression:
[tex]\[
5 \sqrt{3} - 3 \sqrt{3} - 7 \sqrt{3}
\][/tex]
Combine the terms with [tex]\(\sqrt{3}\)[/tex]:
[tex]\[
(5 - 3 - 7) \sqrt{3} = -5 \sqrt{3}
\][/tex]
The result is:
[tex]\[
-5 \sqrt{3} \approx -8.660254037844387
\][/tex]
### Summary of Results:
1. [tex]\(5 \sqrt{2} + 7 \sqrt{2} - 4 \sqrt{2} = 8 \sqrt{2} \approx 11.313708498984763\)[/tex]
2. [tex]\((4 + 3 \sqrt{3})(4 + 4 \sqrt{3}) = 52 + 28 \sqrt{3} \approx 80.08587988004845\)[/tex]
3. [tex]\(5 \sqrt{3} - \sqrt{27} - 7 \sqrt{3} = -5 \sqrt{3} \approx -8.660254037844387\)[/tex]
