Herleid.
1p
\(\sqrt{1\frac{9}{16}}\)
○
\(\sqrt{1\frac{9}{16}} = \sqrt{\frac{25}{16}} = {\sqrt{25} \over \sqrt{16}} = \frac{5}{4} = 1\frac{1}{4} \text{.}\)
1p
Herleid.
1p
\(\sqrt{1\frac{7}{36}}\)
○
\(\sqrt{1\frac{7}{36}} = \sqrt{\frac{43}{36}} = {\sqrt{43} \over \sqrt{36}} = {\sqrt{43} \over 6} = \frac{1}{6} \sqrt{43} \text{.}\)
1p
Herleid.
1p
\(\sqrt{\frac{36}{43}}\)
○
\(\sqrt{\frac{36}{43}} = {\sqrt{36} \over \sqrt{43}} = {6 \over \sqrt{43}} ⋅ {\sqrt{43} \over \sqrt{43}} = {6 \sqrt{43} \over 43} = \frac{6}{43} \sqrt{43} \text{.}\)
1p
Herleid.
1p
\(\sqrt{6\frac{1}{3}}\)
○
\(\sqrt{6\frac{1}{3}} = \sqrt{\frac{19}{3}} = {\sqrt{19} \over \sqrt{3}} ⋅ {\sqrt{3} \over \sqrt{3}} = {\sqrt{57} \over 3} = \frac{1}{3} \sqrt{57} \text{.}\)
1p
Herleid.
1p
\({28 \sqrt{84} \over 4 \sqrt{3}}\)
○
\({28 \sqrt{84} \over 4 \sqrt{3}} = {28 \over 4} ⋅ {\sqrt{84} \over \sqrt{3}} = 7 \sqrt{28} = 7 ⋅ \sqrt{4} ⋅ \sqrt{7} = 7 ⋅ 2 ⋅ \sqrt{7} = 14 \sqrt{7}\)
1p
Herleid.
1p
\(\sqrt{45}\)
○
\(\sqrt{45} = \sqrt{9} ⋅ \sqrt{5} = 3 \sqrt{5} \text{.}\)
1p
Herleid.
1p
\(7 \sqrt{500}\)
○
\(7 \sqrt{500} = 7 ⋅ \sqrt{100} ⋅ \sqrt{5} = 7 ⋅ 10 ⋅ \sqrt{5} = 70 \sqrt{5} \text{.}\)
1p
Herleid.
2p
\(\sqrt{200} + \sqrt{8}\)
○
\(\sqrt{200} + \sqrt{8} = \sqrt{100} ⋅ \sqrt{2} + \sqrt{4} ⋅ \sqrt{2} = 10 \sqrt{2} + 2 \sqrt{2} \text{.}\)
1p
○
\(10 \sqrt{2} + 2 \sqrt{2} = 12 \sqrt{2} \text{.}\)
1p
Herleid.
2p
\(5 \sqrt{50} - 2 \sqrt{18}\)
○
\(5 \sqrt{50} - 2 \sqrt{18} = 5 ⋅ \sqrt{25} ⋅ \sqrt{2} - 2 ⋅ \sqrt{9} ⋅ \sqrt{2} \text{.}\)
1p
○
\(5 ⋅ 5 ⋅ \sqrt{2} - 2 ⋅ 3 ⋅ \sqrt{2} = 25 \sqrt{2} - 6 \sqrt{2} = 19 \sqrt{2} \text{.}\)
1p
Herleid.
1p
\({-4 \over 3 + \sqrt{2}}\)
○
\({-4 \over 3 + \sqrt{2}} = {-4 \over 3 + \sqrt{2}} ⋅ {3 - \sqrt{2} \over 3 - \sqrt{2}}\)
\(\text{} = {-4 (3 + \sqrt{2}) \over 9 - 2}\)
\(\text{} = -\frac{4}{7} (3 + \sqrt{2})\)
\(\text{} = -1\frac{5}{7} - \frac{4}{7} \sqrt{2}\)
1p
Herleid.
1p
\({\sqrt{3} \over \sqrt{6} - \sqrt{2}}\)
○
\({\sqrt{3} \over \sqrt{6} - \sqrt{2}} = {\sqrt{3} \over \sqrt{6} - \sqrt{2}} ⋅ {\sqrt{6} + \sqrt{2} \over \sqrt{6} + \sqrt{2}}\)
\(\text{} = {\sqrt{3} (\sqrt{6} + \sqrt{2}) \over 6 - 2}\)
\(\text{} = \frac{1}{4} \sqrt{3} (\sqrt{6} + \sqrt{2})\)
\(\text{} = \frac{1}{4} \sqrt{18} + \frac{1}{4} \sqrt{6}\)
\(\text{} = \frac{3}{4} \sqrt{2} + \frac{1}{4} \sqrt{6}\)
1p
Herleid.
1p
\(3 \sqrt{6} ⋅ 4 \sqrt{3}\)
○
\(3 \sqrt{6} ⋅ 4 \sqrt{3} = 12 \sqrt{18} = 12 ⋅ \sqrt{9} ⋅ \sqrt{2} = 12 ⋅ 3 ⋅ \sqrt{2} = 36 \sqrt{2}\)
1p
Herleid.
1p
\({4 \over 9 \sqrt{5}}\)
○
\({4 \over 9 \sqrt{5}} = {4 \over 9 \sqrt{5}} ⋅ {\sqrt{5} \over \sqrt{5}} = {4 \sqrt{5} \over 9 ⋅ 5} = \frac{4}{45} \sqrt{5} \text{.}\)
1p