(2) (√1
15 - √5 )^2
Answers
Answer:
We know that,
( \sqrt{a} + \sqrt{b} ) {}^{2} = a + b + 2 \sqrt{ab}(
a
+
b
)
2
=a+b+2
ab
Now,
\sqrt{5} + 2 \sqrt{6} = a + b + 2 \sqrt{a}
5
+2
6
=a+b+2
a
\star \: a + b = \sqrt{5} \rightarrow \: ab = 6 \rightarrow \: a = 3 \: and \: b = 2⋆a+b=
5
→ab=6→a=3andb=2
5 + 2 \sqrt{6} = 3 + 2 + 2 \sqrt{} (3 \times 2)5+2
6
=3+2+2
(3×2)
5 + 2 \sqrt{6} = ( \sqrt{3} + \sqrt{2} ) {}^{2}5+2
6
=(
3
+
2
)
2
\therefore \: 5 + 2 \sqrt{6} = ( \sqrt{3} + \sqrt{2} )∴5+2
6
=(
3
+
2
)
Similarly,
\sqrt{8} - 2 \sqrt{15 } = a + b - 2 \sqrt{ab}
8
−2
15
=a+b−2
ab
\star \: a + b = 8 \rightarrow \: ab = 15 \rightarrow \: a = 5 \: and \: b = 3⋆a+b=8→ab=15→a=5andb=3
\sqrt{8} - 2 \sqrt{15} = 5 + 3 - 2 \sqrt{} (5 \times 3)
8
−2
15
=5+3−2
(5×3)
\sqrt{8} - 2 \sqrt{15 } = ( \sqrt{5} - \sqrt{3} ) {}^{2}
8
−2
15
=(
5
−
3
)
2
\sqrt{8} - 2 \sqrt{15} = ( \sqrt{5} - \sqrt{3} )
8
−2
15
=(
5
−
3
)
According to the question:
( \sqrt{5} + 2 \sqrt{6} )+ (\sqrt{8} - 2 \sqrt{15} )(
5
+2
6
)+(
8
−2
15
)
= \sqrt{3} + \sqrt{2} + \sqrt{5} - \sqrt{3}=
3
+
2
+
5
−
3
= \sqrt{2} + \sqrt{5}=
2
+
5