Herleid.
1p
\(\sqrt{\frac{1}{16}}\)
○
\(\sqrt{\frac{1}{16}} = {\sqrt{1} \over \sqrt{16}} = \frac{1}{4} \text{.}\)
1p
Herleid.
1p
\(\sqrt{11\frac{3}{4}}\)
○
\(\sqrt{11\frac{3}{4}} = \sqrt{\frac{47}{4}} = {\sqrt{47} \over \sqrt{4}} = {\sqrt{47} \over 2} = \frac{1}{2} \sqrt{47} \text{.}\)
1p
Herleid.
1p
\(\sqrt{9\frac{1}{11}}\)
○
\(\sqrt{9\frac{1}{11}} = \sqrt{\frac{100}{11}} = {\sqrt{100} \over \sqrt{11}} = {10 \over \sqrt{11}} ⋅ {\sqrt{11} \over \sqrt{11}} = {10 \sqrt{11} \over 11} = \frac{10}{11} \sqrt{11} \text{.}\)
1p
Herleid.
1p
\(\sqrt{22\frac{1}{2}}\)
○
\(\sqrt{22\frac{1}{2}} = \sqrt{\frac{45}{2}} = {\sqrt{45} \over \sqrt{2}} ⋅ {\sqrt{2} \over \sqrt{2}} = {\sqrt{90} \over 2} = \frac{1}{2} \sqrt{90} = \frac{1}{2} ⋅ 3 ⋅ \sqrt{10} = 1\frac{1}{2} \sqrt{10} \text{.}\)
1p
Herleid.
1p
\({5 \sqrt{140} \over \sqrt{7}}\)
○
\({5 \sqrt{140} \over \sqrt{7}} = 5 ⋅ {\sqrt{140} \over \sqrt{7}} = 5 \sqrt{20} = 5 ⋅ \sqrt{4} ⋅ \sqrt{5} = 5 ⋅ 2 ⋅ \sqrt{5} = 10 \sqrt{5}\)
1p
Herleid.
1p
\(\sqrt{75}\)
○
\(\sqrt{75} = \sqrt{25} ⋅ \sqrt{3} = 5 \sqrt{3} \text{.}\)
1p
Herleid.
1p
\(-4 \sqrt{18}\)
○
\(-4 \sqrt{18} = -4 ⋅ \sqrt{9} ⋅ \sqrt{2} = -4 ⋅ 3 ⋅ \sqrt{2} = -12 \sqrt{2} \text{.}\)
1p
Herleid.
2p
\(\sqrt{32} + \sqrt{200}\)
○
\(\sqrt{32} + \sqrt{200} = \sqrt{16} ⋅ \sqrt{2} + \sqrt{100} ⋅ \sqrt{2} = 4 \sqrt{2} + 10 \sqrt{2} \text{.}\)
1p
○
\(4 \sqrt{2} + 10 \sqrt{2} = 14 \sqrt{2} \text{.}\)
1p
Herleid.
2p
\(3 \sqrt{20} + 5 \sqrt{45}\)
○
\(3 \sqrt{20} + 5 \sqrt{45} = 3 ⋅ \sqrt{4} ⋅ \sqrt{5} + 5 ⋅ \sqrt{9} ⋅ \sqrt{5} \text{.}\)
1p
○
\(3 ⋅ 2 ⋅ \sqrt{5} + 5 ⋅ 3 ⋅ \sqrt{5} = 6 \sqrt{5} + 15 \sqrt{5} = 21 \sqrt{5} \text{.}\)
1p
Herleid.
1p
\({3 \over 4 + \sqrt{5}}\)
○
\({3 \over 4 + \sqrt{5}} = {3 \over 4 + \sqrt{5}} ⋅ {4 - \sqrt{5} \over 4 - \sqrt{5}}\)
\(\text{} = {3 (4 + \sqrt{5}) \over 16 - 5}\)
\(\text{} = \frac{3}{11} (4 + \sqrt{5})\)
\(\text{} = 1\frac{1}{11} + \frac{3}{11} \sqrt{5}\)
1p
Herleid.
1p
\({2 \sqrt{6} \over \sqrt{3} + \sqrt{5}}\)
○
\({2 \sqrt{6} \over \sqrt{3} + \sqrt{5}} = {2 \sqrt{6} \over \sqrt{3} + \sqrt{5}} ⋅ {\sqrt{3} - \sqrt{5} \over \sqrt{3} - \sqrt{5}}\)
\(\text{} = {2 \sqrt{6} (\sqrt{3} - \sqrt{5}) \over 3 - 5}\)
\(\text{} = -\sqrt{6} (\sqrt{3} - \sqrt{5})\)
\(\text{} = -\sqrt{18} + \sqrt{30}\)
\(\text{} = -3 \sqrt{2} + \sqrt{30}\)
1p
Herleid.
1p
\(4 \sqrt{15} ⋅ 5 \sqrt{6}\)
○
\(4 \sqrt{15} ⋅ 5 \sqrt{6} = 20 \sqrt{90} = 20 ⋅ \sqrt{9} ⋅ \sqrt{10} = 20 ⋅ 3 ⋅ \sqrt{10} = 60 \sqrt{10}\)
1p
Herleid.
1p
\({5 \over 3 \sqrt{3}}\)
○
\({5 \over 3 \sqrt{3}} = {5 \over 3 \sqrt{3}} ⋅ {\sqrt{3} \over \sqrt{3}} = {5 \sqrt{3} \over 3 ⋅ 3} = \frac{5}{9} \sqrt{3} \text{.}\)
1p