Question

If 1/m+1/n=-3 an mn=-1/54 then find the value of 1/m^3+1/n^3

Answer 1

Given,

$mn \: = \frac{ - 1}{54} .............(1)$

Also,

$\frac{1}{m} + \frac{1}{n} = - 3$

$\frac{m \: + \: n}{mn} = - 3$

$m \: + \: n = - 3mn$

Substituting the value of 'mn' from equation (1),

$m \: + \: n = (- 3)(\frac{ - 1}{54} )$

$m \: + \: n = \frac{3}{54} \: ........(2)$

$now \: the \: value \: of \: \frac{1}{ {m}^{3} } + \frac{1}{ {n}^{3} }$

$= \frac{{m}^{3} + {n}^{3}}{{(mn)}^{3} }$

From above image we can write m^3 + n^3 as,

$= \frac{{(m \: + n)}^{3} \: - 3mn(m \: + \: n)}{{(mn)}^{3} }$

Now substituting the value of mn and m+n from (1) and (2), we get,

$= \frac{{( \frac{3}{54} )}^{3} \: - \: 3( \frac{ - 1}{54} )(\frac{3}{54} )}{{( \frac{ - 1}{54} )}^{3} }$

$= \frac{{( \frac{3}{54} )}^{3} \: + \: {( \frac{ 3}{54} )}^{2} }{{( \frac{ - 1}{54} )}^{3} }$

$=\frac{ {3}^{3} + ({3}^{2} )(54)}{( - 1)}$

$= ( - 1)({3}^{2})(3 + 54)$

$= ( - 1)(9)(57)$

$= ( - 1)(513)$

$= ( -513)$


Answer: -513

Answer 2

Answer:

Step-by-step explanation: