simplify [4+√5/4-√5+4-√5/4+√5]
Answers
Step-by-step explanation:
Given:
\mathsf{\dfrac{4+\sqrt{5}}{4-\sqrt{5}}+\dfrac{4-\sqrt{5}}{4+\sqrt{5}}}
4−
5
4+
5
+
4+
5
4−
5
\underline{\textbf{To simplify:}}
To simplify:
\mathsf{\dfrac{4+\sqrt{5}}{4-\sqrt{5}}+\dfrac{4-\sqrt{5}}{4+\sqrt{5}}}
4−
5
4+
5
+
4+
5
4−
5
\underline{\textbf{Solution:}}
Solution:
\mathsf{Consider,}Consider,
\mathsf{\dfrac{4+\sqrt{5}}{4-\sqrt{5}}+\dfrac{4-\sqrt{5}}{4+\sqrt{5}}}
4−
5
4+
5
+
4+
5
4−
5
\textsf{This can be written as,}This can be written as,
\mathsf{=\dfrac{(4+\sqrt{5})(4+\sqrt{5})+(4-\sqrt{5})(4-\sqrt{5})}{(4-\sqrt{5})(4+\sqrt{5})}}=
(4−
5
)(4+
5
)
(4+
5
)(4+
5
)+(4−
5
)(4−
5
)
\mathsf{=\dfrac{(4+\sqrt{5})^2+(4-\sqrt{5})^2}{(4-\sqrt{5})(4+\sqrt{5})}}=
(4−
5
)(4+
5
)
(4+
5
)
2
+(4−
5
)
2
\textsf{Using the following identities,}Using the following identities,
\mathsf{(a-b)(a+b)=a^2-b^2}(a−b)(a+b)=a
2
−b
2
\mathsf{(a+b)^2=a^2+b^2+2ab}(a+b)
2
=a
2
+b
2
+2ab
\mathsf{(a-b)^2=a^2+b^2-2ab}(a−b)
2
=a
2
+b
2
−2ab
\mathsf{=\dfrac{4^2+(\sqrt{5})^2+8\sqrt{5}+4^2+(\sqrt{5})^2-8\sqrt{5}}{4^2-(\sqrt{5})^2}}=
4
2
−(
5
)
2
4
2
+(
5
)
2
+8
5
+4
2
+(
5
)
2
−8
5
\mathsf{=\dfrac{16+5+16+5}{16-5}}=
16−5
16+5+16+5
\mathsf{=\dfrac{42}{11}}=
11
42