Question

if x = 4+ √15 then the value of x^3 + 1/x^3 is
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Answer 1

Answer:

488

Step-by-step explanation:

Given x = 4+ √15

So, 1/x = 1/4+ √15

           = 4 - √15/ (4 - √15)(/4+ √15)

           = 4 - √15/(16 - 15)

           = 4 - √15

We know that $(a + b)^{3}$ + $(a - b)^{3}$ = 2($a^{3}$+$3ab^2$)

So, $x^{3} + 1/x^{3}$

= $(4+\sqrt{15} )^3 + (4-\sqrt{15} )^3 = 2(4^3 + 3*4*\sqrt{15}^2)$

= 2(64+180)

= 128 + 360

= 488

Answer 2

Given,

$x=4+\sqrt{15}$

We have to find the value of $x^{3}+\frac{1}{x^3}$

First we have to find the value of $\frac{1}{x}$

$x= 4+\sqrt{15}$

$\frac{1}{x}=\frac{1}{4+\sqrt{15} }$

Multiply and divided by $4-\sqrt{15}$ with numerator and denominator

$= \frac{1}{4+\sqrt{15} }\times\frac{4-\sqrt{15} }{4-\sqrt{15} }$

$= \frac{4+\sqrt{15} }{16-15}$

$\frac{1}{x} = {4-\sqrt{15} }$

Now, substitute $x$ and $\frac{1}{x}$ values  $x^{3}+\frac{1}{x^3}$

$=(4+\sqrt{15} )^{3}+(4-\sqrt{15})^{3}$

We know that,

$(a+b)^{3}+(a-b)^{3}= 2(a^{3}+3ab^{2} )$

$(4+\sqrt{15} )^{3} +(4-\sqrt{15} )^{3}= 2(4^{3}+3\times4\times(\sqrt{15} )^{2} )$

$= 2(64+180)$

$= 2(224)$

$= 488$

Therefor, the answer is $488$.