Question

If x = √3+√2/ √3-√2 , find the value of x^2 + 2/x^2 and x^4 + 1/x^4 .​

Answer 1

Answer:

Step-by-step explanation:

Step-by-step explanation:

We have,

x=3+2\sqrt{2}

To find, the value of x^{4} +\dfrac{1}{x^{4}} =?

∴ \dfrac{1}{x} =\dfrac{1}{3+2\sqrt{2}}

Rationalising, we get

\dfrac{1}{x} =\dfrac{1}{3+2\sqrt{2}}\times \dfrac{3-2\sqrt{2}}{3-2\sqrt{2}}

=\dfrac{3-2\sqrt{2}}{3^2-(2\sqrt{2})^2}=\dfrac{3-2\sqrt{2}}{9-8}

\dfrac{1}{x} =3-2\sqrt{2}

∴ (x+\frac{1}{x})^{2}=x^{2}+(\dfrac{1}{x})^{2}+2.x.\dfrac{1}{x}

⇒ (3+2\sqrt{2}+3-2\sqrt{2})^{2}=x^{2}+\dfrac{1}{x^{2}} +2

⇒ x^{2}+\dfrac{1}{x^{2}} =36-2=34     .....(1)

Again squaring (1), we get

(x^{2}+\dfrac{1}{x^{2}} )^{2} =34^{2}

x^{4}+\dfrac{1}{x^{4}} +2=1156

⇒ x^{4}+\dfrac{1}{x^{4}} =1156-2=1154  

Hence, x^{4}+\dfrac{1}{x^{4}}=1154Step-by-step explanation:

We have,

x=3+2\sqrt{2}

To find, the value of x^{4} +\dfrac{1}{x^{4}} =?

∴ \dfrac{1}{x} =\dfrac{1}{3+2\sqrt{2}}

Rationalising, we get

\dfrac{1}{x} =\dfrac{1}{3+2\sqrt{2}}\times \dfrac{3-2\sqrt{2}}{3-2\sqrt{2}}

=\dfrac{3-2\sqrt{2}}{3^2-(2\sqrt{2})^2}=\dfrac{3-2\sqrt{2}}{9-8}

\dfrac{1}{x} =3-2\sqrt{2}

∴ (x+\frac{1}{x})^{2}=x^{2}+(\dfrac{1}{x})^{2}+2.x.\dfrac{1}{x}

⇒ (3+2\sqrt{2}+3-2\sqrt{2})^{2}=x^{2}+\dfrac{1}{x^{2}} +2

⇒ x^{2}+\dfrac{1}{x^{2}} =36-2=34     .....(1)

Again squaring (1), we get

(x^{2}+\dfrac{1}{x^{2}} )^{2} =34^{2}

x^{4}+\dfrac{1}{x^{4}} +2=1156

⇒ x^{4}+\dfrac{1}{x^{4}} =1156-2=1154  

Hence, x^{4}+\dfrac{1}{x^{4}}=1154Step-by-step explanation:

We have,

x=3+2\sqrt{2}

To find, the value of x^{4} +\dfrac{1}{x^{4}} =?

∴ \dfrac{1}{x} =\dfrac{1}{3+2\sqrt{2}}

Rationalising, we get

\dfrac{1}{x} =\dfrac{1}{3+2\sqrt{2}}\times \dfrac{3-2\sqrt{2}}{3-2\sqrt{2}}

=\dfrac{3-2\sqrt{2}}{3^2-(2\sqrt{2})^2}=\dfrac{3-2\sqrt{2}}{9-8}

\dfrac{1}{x} =3-2\sqrt{2}

∴ (x+\frac{1}{x})^{2}=x^{2}+(\dfrac{1}{x})^{2}+2.x.\dfrac{1}{x}

⇒ (3+2\sqrt{2}+3-2\sqrt{2})^{2}=x^{2}+\dfrac{1}{x^{2}} +2

⇒ x^{2}+\dfrac{1}{x^{2}} =36-2=34     .....(1)

Again squaring (1), we get

(x^{2}+\dfrac{1}{x^{2}} )^{2} =34^{2}

x^{4}+\dfrac{1}{x^{4}} +2=1156

⇒ x^{4}+\dfrac{1}{x^{4}} =1156-2=1154  

Hence, x^{4}+\dfrac{1}{x^{4}}=1154

Answer 2

Answer:

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