Answer :
Sure, let's work through this problem step by step.
### Part 1: Simplifying [tex]\(\sqrt{80} + \sqrt{20} - \sqrt{45}\)[/tex]
1. Calculate [tex]\(\sqrt{80}\)[/tex]:
[tex]\[ \sqrt{80} \approx 8.94427190999916 \][/tex]
2. Calculate [tex]\(\sqrt{20}\)[/tex]:
[tex]\[ \sqrt{20} \approx 4.47213595499958 \][/tex]
3. Calculate [tex]\(\sqrt{45}\)[/tex]:
[tex]\[ \sqrt{45} \approx 6.708203932499369 \][/tex]
4. Form the expression [tex]\(\sqrt{80} + \sqrt{20} - \sqrt{45}\)[/tex]:
[tex]\[ \sqrt{80} + \sqrt{20} - \sqrt{45} \approx 8.94427190999916 + 4.47213595499958 - 6.708203932499369 \][/tex]
5. Simplify the expression:
[tex]\[ \sqrt{80} + \sqrt{20} - \sqrt{45} \approx 6.708203932499369 \][/tex]
So, the simplified result of [tex]\(\sqrt{80} + \sqrt{20} - \sqrt{45}\)[/tex] is approximately [tex]\(6.708203932499369\)[/tex].
### Part 2: Finding the positive square root of [tex]\(7 + 2 \sqrt{6}\)[/tex]
1. Calculate [tex]\(2 \sqrt{6}\)[/tex]:
[tex]\[ 2 \sqrt{6} \approx 4.898979485566356 \][/tex]
2. Form the expression [tex]\(7 + 2 \sqrt{6}\)[/tex]:
[tex]\[ 7 + 2 \sqrt{6} \approx 7 + 4.898979485566356 = 11.898979485566356 \][/tex]
3. Calculate the positive square root of [tex]\(11.898979485566356\)[/tex]:
[tex]\[ \sqrt{11.898979485566356} \approx 3.449489742783178 \][/tex]
So, the positive square root of [tex]\(7 + 2 \sqrt{6}\)[/tex] is approximately [tex]\(3.449489742783178\)[/tex].
I hope this breakdown helps you understand how the problem is solved!
### Part 1: Simplifying [tex]\(\sqrt{80} + \sqrt{20} - \sqrt{45}\)[/tex]
1. Calculate [tex]\(\sqrt{80}\)[/tex]:
[tex]\[ \sqrt{80} \approx 8.94427190999916 \][/tex]
2. Calculate [tex]\(\sqrt{20}\)[/tex]:
[tex]\[ \sqrt{20} \approx 4.47213595499958 \][/tex]
3. Calculate [tex]\(\sqrt{45}\)[/tex]:
[tex]\[ \sqrt{45} \approx 6.708203932499369 \][/tex]
4. Form the expression [tex]\(\sqrt{80} + \sqrt{20} - \sqrt{45}\)[/tex]:
[tex]\[ \sqrt{80} + \sqrt{20} - \sqrt{45} \approx 8.94427190999916 + 4.47213595499958 - 6.708203932499369 \][/tex]
5. Simplify the expression:
[tex]\[ \sqrt{80} + \sqrt{20} - \sqrt{45} \approx 6.708203932499369 \][/tex]
So, the simplified result of [tex]\(\sqrt{80} + \sqrt{20} - \sqrt{45}\)[/tex] is approximately [tex]\(6.708203932499369\)[/tex].
### Part 2: Finding the positive square root of [tex]\(7 + 2 \sqrt{6}\)[/tex]
1. Calculate [tex]\(2 \sqrt{6}\)[/tex]:
[tex]\[ 2 \sqrt{6} \approx 4.898979485566356 \][/tex]
2. Form the expression [tex]\(7 + 2 \sqrt{6}\)[/tex]:
[tex]\[ 7 + 2 \sqrt{6} \approx 7 + 4.898979485566356 = 11.898979485566356 \][/tex]
3. Calculate the positive square root of [tex]\(11.898979485566356\)[/tex]:
[tex]\[ \sqrt{11.898979485566356} \approx 3.449489742783178 \][/tex]
So, the positive square root of [tex]\(7 + 2 \sqrt{6}\)[/tex] is approximately [tex]\(3.449489742783178\)[/tex].
I hope this breakdown helps you understand how the problem is solved!
