Question

If
$x + \frac{1}{x} = 5$
then find the value of
${x}^{9} + \frac{1}{{x}^{9} }$
Don't write anything useless else I'll have to report your answer​

Answer 1

Step-by-step explanation:

Given :-

x+(1/x) = 5

To find :-

Find the value of x⁹+(1/x⁹) ?

Solution :-

Given that :

x+(1/x) = 5 ------------(1)

On cubing both sides

[x+(1/x)]³ = 5³

=> [x+(1/x)]³ = 125

We know that

(a+b)³ = a³+b³+3ab(a+b)

Where a = x and b = 1/x

Now,

=>x³+(1/x)³+3(x)(1/x)[x+(1/x)] = 125

=> x³+(1/x³) +3(x/x)[x+(1/x)] = 125

=> x³+(1/x³) +3(1)[x+(1/x)] = 125

=>x³+(1/x³) +3[x+(1/x)] = 125

=> x³+(1/x³)+3(5) = 125 (from (1))

=> x³+(1/x³)+ 15= 125

=> x³+(1/x³) = 125-15

=> x³+(1/x³) = 110 ----------(2)

On cubing both sides then

=> [x³+(1/x³)]³ = (110)³

We know that

(a+b)³ = a³+b³+3ab(a+b)

Where a = x³ and b = 1/x³

Now,

=>(x³)³+(1/x³)³+3(x³)(1/x³)[x³+(1/x³)] = 1331000

=> x⁹+(1/x⁹)+3(x³/x³)[x³+(1/x³)] = 1331000

=> x⁹+(1/x⁹)+3(1)[x³+(1/x³)] = 1331000

=> x⁹+(1/x⁹)+3[x³+(1/x³)] = 1331000

=> x⁹+(1/x⁹)+3(110) = 1331000

=> x⁹+(1/x⁹) +330 = 1331000

=> x⁹+(1/x⁹) = 1331000-330

=> x⁹+(1/x⁹) = 1330670

Answer :-

The value of x⁹+(1/x⁹) for the given problem is 1330670

Used formulae:-

• (a+b)³ = a³+b³+3ab(a+b)
• (a^m)^n = a^(mn)