4√2/√15-3√2 + 3√5/√10-√3 + 5√5/√6+√5
Answers
Heya !!!
\begin{gathered} \frac{4 \sqrt{2} }{ \sqrt{15} - 3 \sqrt{2} } + \frac{3 \sqrt{5} }{ \sqrt{10} - \sqrt{3} } + \frac{5 \sqrt{3} }{ \sqrt{6} + \sqrt{5} } \\\end{gathered}
15
−3
2
4
2
+
10
−
3
3
5
+
6
+
5
5
3
On rationalizing the denominator we get,
\begin{gathered} = \frac{4 \sqrt{2} }{ \sqrt{15} - 3 \sqrt{2} } \times \frac{ \sqrt{15} + 3 \sqrt{2} }{ \sqrt{15} + 3 \sqrt{2} } + \frac{3 \sqrt{5} }{ \sqrt{10} - \sqrt{3} } \times \frac{ \sqrt{10} + \sqrt{3} }{ \sqrt{10} + \sqrt{3} } + \frac{5 \sqrt{3} }{ \sqrt{6} + \sqrt{5} } \times \frac{ \sqrt{6} - \sqrt{5} }{ \sqrt{6} - \sqrt{5} } \\ \\ = \frac{4 \sqrt{2} ( \sqrt{15} + 3 \sqrt{2} )}{ {( \sqrt{15}) }^{2} - {(3 \sqrt{2} )}^{2} } + \frac{3 \sqrt{5}( \sqrt{10} + \sqrt{3} )}{ {( \sqrt{10}) }^{2} - {( \sqrt{3} )}^{2} } + \frac{5 \sqrt{3} ( \sqrt{6} - \sqrt{5}) }{ {( \sqrt{6}) }^{2} - {( \sqrt{5}) }^{2} } \\ \\ = \frac{4 \sqrt{30} + 24 }{15 - 18} + \frac{3 \sqrt{50} + 3 \sqrt{15} }{10 - 3} + \frac{5 \sqrt{18} - 5 \sqrt{15} }{6 - 5} \\ \\ = \frac{4 \sqrt{30} + 24}{ - 3} + \frac{3 \sqrt{2 \times 5 \times 5} + 3 \sqrt{15} }{7} + 5 \sqrt{3 \times 3 \times 2} - 5 \sqrt{15} \\ \\ = \frac{ - 4 \sqrt{30} - 24}{3} + \frac{15 \sqrt{2} + 3 \sqrt{15} }{7} + 15 \sqrt{2} - 5 \sqrt{15} \\ \\ = \frac{ - 4 \sqrt{30} \times 7 - 24 \times 7 + 15 \sqrt{2} \times 3 + 3 \sqrt{15} \times 3 + 15 \sqrt{2} \times 21 - 5 \sqrt{15} \times 21}{21} \\ \\ = \frac{ - 28 \sqrt{30} - 168 + 45 \sqrt{2} + 9 \sqrt{15} + 315 \sqrt{2} - 105 \sqrt{15} }{21} \\ \\ = \frac{ - 28 \sqrt{30} - 168 + 360 \sqrt{2} - 96 \sqrt{15} }{21} \end{gathered}
=
15
−3
2
4
2
×
15
+3
2
15
+3
2
+
10
−
3
3
5
×
10
+
3
10
+
3
+
6
+
5
5
3
×
6
−
5
6
−
5
=
(
15
)
2
−(3
2
)
2
4
2
(
15
+3
2
)
+
(
10
)
2
−(
3
)
2
3
5
(
10
+
3
)
+
(
6
)
2
−(
5
)
2
5
3
(
6
−
5
)
=
15−18
4
30
+24
+
10−3
3
50
+3
15
+
6−5
5
18
−5
15
=
−3
4
30
+24
+
7
3
2×5×5
+3
15
+5
3×3×2
−5
15
=
3
−4
30
−24
+
7
15
2
+3
15
+15
2
−5
15
=
21
−4
30
×7−24×7+15
2
×3+3
15
×3+15
2
×21−5
15
×21
=
21
−28
30
−168+45
2
+9
15
+315
2
−105
15
=
21
−28
30
−168+360
2
−96
15