Question

If x=3+2√2,then find the value of (x-1/(x))3​

Answer 1

$(x + \frac{1}{x} {)}^{3}=198$

Step-by-step explanation:

Given Information -

$x = 3 + 2 \sqrt{2}$

ㅤ

To Find-

$(x + \frac{1}{x} {)}^{3}$

ㅤ

Solution -

Step-1=Find the value of Reciprocal of 'x' -

• $\color{orange} {x = 3 + 2 \sqrt{2}}$

• $\color{orange} {\frac{1}{x} = \frac{1}{3 + 2 \sqrt{2} }}$

• $\color{orange} {\frac{1}{3 + 2 \sqrt{2} } \times \frac{3 - 2 \sqrt{2} }{3 - 2 \sqrt{2} }}$

• $\color{orange} {\frac{3 - 2 \sqrt{2} }{(3 + 2 \sqrt{2} )(3 - 2 \sqrt{2} )}}$

• $\color{orange} { \frac{3 - 2 \sqrt{2} }{(3 + 2 \sqrt{2} )(3 - 2 \sqrt{2} )}}$

• $\color{orange} {\frac{3 - 2 \sqrt{2} }{(3 {)}^{2} - (2 \sqrt{2} {)}^{2} }}$

• $\color{orange} {\frac{3 - 2 \sqrt{2} }{9 - 8}}$

• $\sf\red {\frac{1}{x} = 3 - 2 \sqrt{2}}$

ㅤ✎﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏

Step-2= Find the value of $\underline{x + \frac{1}{x}}$ -

• $\color{purple} {x + \frac{1}{x}}$

• $\color{purple}{3 + 2 \sqrt{2} + 3 - 2 \sqrt{2}}$

• $\color{purple} {3 + 2 \sqrt{2} + 3 - 2 \sqrt{2}}$

• $\color{purple} {3 + 3}$

• $\sf\blue{ 6}$

ㅤ✎﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏

Step-3=Find The value of $\underline{(x + \frac{1}{x} {)}^{3}}$ -

• $\color{green}{(x + \frac{1}{x} {)}^{3} = (x + \frac{1}{x} {)}^{3} - 3(x + \frac{1}{x} )}$

• $\color{green}{{6}^{3} - (3 \times 6)}$

• $\color{green}{(6 \times 6 \times 6)- (3 \times 6)}$

• $\color{green}{216 - 18}$

• $\huge\sf\color{lime}{198}$

Answer 2

Answer:

(x + \frac{1}{x} {)}^{3}=198(x+

x

1

)

3

=198

Step-by-step explanation:

Given Information -

x = 3 + 2 \sqrt{2}x=3+2

2

ㅤ

To Find-

(x + \frac{1}{x} {)}^{3}(x+

x

1

)

3

ㅤ

Solution -

Step-1=Find the value of Reciprocal of 'x' -

\color{orange} {x = 3 + 2 \sqrt{2}}x=3+2

2

\color{orange} {\frac{1}{x} = \frac{1}{3 + 2 \sqrt{2} }}

x

1

=

3+2

2

1

\color{orange} {\frac{1}{3 + 2 \sqrt{2} } \times \frac{3 - 2 \sqrt{2} }{3 - 2 \sqrt{2} }}

3+2

2

1

×

3−2

2

3−2

2

\color{orange} {\frac{3 - 2 \sqrt{2} }{(3 + 2 \sqrt{2} )(3 - 2 \sqrt{2} )}}

(3+2

2

)(3−2

2

)

3−2

2

\color{orange} { \frac{3 - 2 \sqrt{2} }{(3 + 2 \sqrt{2} )(3 - 2 \sqrt{2} )}}

(3+2

2

)(3−2

2

)

3−2

2

\color{orange} {\frac{3 - 2 \sqrt{2} }{(3 {)}^{2} - (2 \sqrt{2} {)}^{2} }}

(3)

2

−(2

2

)

2

3−2

2

\color{orange} {\frac{3 - 2 \sqrt{2} }{9 - 8}}

9−8

3−2

2

\sf\red {\frac{1}{x} = 3 - 2 \sqrt{2}}

x

1

=3−2

2

ㅤ✎﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏

Step-2= Find the value of \underline{x + \frac{1}{x}}

x+

x

1

-

\color{purple} {x + \frac{1}{x}}x+

x

1

\color{purple}{3 + 2 \sqrt{2} + 3 - 2 \sqrt{2}}3+2

2

+3−2

2

\color{purple} {3 + 2 \sqrt{2} + 3 - 2 \sqrt{2}}3+2

2

+3−2

2

\color{purple} {3 + 3}3+3

\sf\blue{ 6}6

ㅤ✎﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏

Step-3=Find The value of \underline{(x + \frac{1}{x} {)}^{3}}

(x+

x

1

)

3

-

\color{green}{(x + \frac{1}{x} {)}^{3} = (x + \frac{1}{x} {)}^{3} - 3(x + \frac{1}{x} )}(x+

x

1

)

3

=(x+

x

1

)

3

−3(x+

x

1

)

\color{green}{{6}^{3} - (3 \times 6)}6

3

−(3×6)

\color{green}{(6 \times 6 \times 6)- (3 \times 6)}(6×6×6)−(3×6)

\color{green}{216 - 18}216−18

\huge\sf\color{lime}{198}198