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Answer:
a. [tex] {x}^{2} + \frac{1}{ {x}^{2} } [/tex]
[tex] {x} + \dfrac{1}{ {x}} = 8[/tex]
[tex]⟹(x + \dfrac{1}{x} )^{2} = (8)^{2} [/tex]
[tex]⟹(x)^{2} + 2 \times x \times \dfrac{1}{x} + ( \dfrac{1}{x} )^{2} = 64[/tex]
[tex]⟹ {x}^{2} + 2 + \dfrac{1}{{x}^{2} } = 64[/tex]
[tex]⟹ {x}^{2} + \dfrac{1}{ {x}^{2} } = 64 - 2[/tex]
Hence , [tex]⟹ {x}^{2} + \dfrac{1}{ {x}^{2} } = 62[/tex]
b . [tex] {x}^{4} + \frac{1}{ {x}^{4} } [/tex]
as,
[tex]{x}^{2} + \dfrac{1}{ {x}^{2} } = 62[/tex]
[tex]⟹[ {x}^{2} + \dfrac{1}{ {x}^{2} } ] ^{2} = (62)^{2} [/tex]
[tex]⟹({x}^{2} )^{2} + 2 \times {x}^{2} \times \dfrac{1}{ {x}^{2} } + [ \dfrac{1}{ {x}^{2} } ] ^{2} = 3844[/tex]
[tex]⟹{x}^{4} + 2 + \dfrac{1}{ {x}^{4} } = 3844[/tex]
[tex]⟹ {x}^{4} + \dfrac{1}{ {x}^{4} } = 3844 - 2 [/tex]
hence, [tex]⟹ {x}^{4} + \dfrac{1}{{x}^{4}} = 3842[/tex]