root whole root 11 + 4 root 7 minus whole root 11 - 4 root 7 is equal to 4 prove
Answers
Answer:
a=
2
9
,b=
2
1
Step-by-step explanation:
LHS=\frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}+\sqrt{7}}LHS=
11
+
7
11
−
7
/* Rationalising the denominator, we get
=\frac{(\sqrt{11}-\sqrt{7})(\sqrt{11}-\sqrt{7})}{(\sqrt{11}+\sqrt{7})(\sqrt{11}-\sqrt{7})}=
(
11
+
7
)(
11
−
7
)
(
11
−
7
)(
11
−
7
)
=\frac{(\sqrt{11}-\sqrt{7})^{2}}{(\sqrt{11})^{2}-(\sqrt{7})^{2}}=
(
11
)
2
−(
7
)
2
(
11
−
7
)
2
=\frac{11+7-2\times \sqrt{11}\times \sqrt{7}}{11-7}=
11−7
11+7−2×
11
×
7
/* By algebraic identities:
i) (a-b)² = a²+b²-2ab
ii) (a+b)(a-b) = a² - b² */
= \frac{18-2\sqrt{77}}{4}=
4
18−2
77
=\frac{2(9-\sqrt{77})}{4}=
4
2(9−
77
)
=\frac{9-\sqrt{77}}{2}=
2
9−
77
\begin{lgathered}=\frac{9}{2}-\frac{\sqrt{77}}{2}\\=RHS\end{lgathered}
=
2
9
−
2
77
=RHS
Therefore,
\begin{lgathered}\frac{9}{2}-\frac{\sqrt{77}}{2}\\=a-b\sqrt{77}\end{lgathered}
2
9
−
2
77
=a−b
77
/* Compare both sides, we get
a=\frac{9}{2},\:b=\frac{1}{2}a=
2
9
,b=
2
1
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