Question

If x+1/x=5 find the value of x^9+1/x^9

Answer 1

GIVEN:-

• $\rm{ x + \dfrac{1}{x} = 5}$

TO FIND :-

• $\rm{ x^9 + \dfrac{1}{x^9}}$

IDENTITIES USED:-

• $\rm{(a+b)^2 = a^3 + b^3 + 3\times{a}\times{b} ( a+b)}$

Now,

$\implies\rm{ (x +\dfrac{1}{x}) = 5 }$

• Cubing it.

$\implies\rm{ (x + \dfrac{1}{x})^3 = x^3 + \dfrac{1}{x^3} + 3\times{x}\times{\dfrac{1}{x}}( x+ \dfrac{1}{x})}$

$\implies\rm{ (5)^3 = x^3 + \dfrac{1}{x^3} + 3\times{x}\times{\dfrac{1}{x}} (5)}$

$\implies\rm{ 125 - 15 =x^3 + \dfrac{1}{x^3}}$

$\implies\rm{ 110 = x^3 + \dfrac{1}{x^3}}$

• Now again Cubing

$\implies\rm{ (x^3 + \dfrac{1}{x^3})^3 = (x^3)^3+ \dfrac{1}{(x^3)^3} + 3\times{x^3}\times{\dfrac{1}{x^3}}( x^3+ \dfrac{1}{x^3})}$

$\implies\rm{ (110)^3 =(x^3)^3\dfrac{1}{(x^3)^3} + 3\times{x^3}\times{\dfrac{1}{x^3}(110)}}$

$\implies\rm{1331000 = x^9 + \dfrac{1}{x^9} + 3\times{110}}$

$\implies\rm{ 1331000-1430 =x^9 + \dfrac{1}{x^9}}$

$\implies\rm{ 1329570 =x^9 + \dfrac{1}{x^9}}$

Answer 2

Step-by-step explanation:

(x+

x

1

)=5

Cubing it.

\implies\rm{ (x + \dfrac{1}{x})^3 = x^3 + \dfrac{1}{x^3} + 3\times{x}\times{\dfrac{1}{x}}( x+ \dfrac{1}{x})}⟹(x+

x

1

)

3

=x

3

+

x

3

1

+3×x×

x

1

(x+

x

1

)

\implies\rm{ (5)^3 = x^3 + \dfrac{1}{x^3} + 3\times{x}\times{\dfrac{1}{x}} (5)}⟹(5)

3

=x

3

+

x

3

1

+3×x×

x

1

(5)

\implies\rm{ 125 - 15 =x^3 + \dfrac{1}{x^3}}⟹125−15=x

3

+

x

3

1

\implies\rm{ 110 = x^3 + \dfrac{1}{x^3}}⟹110=x

3

+

x

3

1

Now again Cubing

\implies\rm{ (x^3 + \dfrac{1}{x^3})^3 = (x^3)^3+ \dfrac{1}{(x^3)^3} + 3\times{x^3}\times{\dfrac{1}{x^3}}( x^3+ \dfrac{1}{x^3})}⟹(x

3

+

x

3

1

)

3

=(x

3

)

3

+

(x

3

)

3

1

+3×x

3

×

x

3

1

(x

3

+

x

3

1

)

\implies\rm{ (110)^3 =(x^3)^3\dfrac{1}{(x^3)^3} + 3\times{x^3}\times{\dfrac{1}{x^3}(110)}}⟹(110)

3

=(x

3

)

3

(x

3

)

3

1

+3×x

3

×

x

3

1

(110)

\implies\rm{1331000 = x^9 + \dfrac{1}{x^9} + 3\times{110}}⟹1331000=x

9

+

x

9

1

+3×110

\implies\rm{ 1331000-1430 =x^9 + \dfrac{1}{x^9}}⟹1331000−1430=x

9

+

x

9

1

\implies\rm{ 1329570 =x^9 + \dfrac{1}{x^9}}⟹1329570=x

9

+

x

9

1