Question

X square + 1 by x square is equals to 51 then find x minus one by x and x cube minus 1 by x cube

Answer 1

it is a simple question
(x^2 =x square )
x^2+1/x^2 = 51
1/x^2=50
x^2=1/50
x=1/7

Answer 2

$\Large{\bold{Hey\:\;\;Mate\;\;\:Here\:\:\:Is\:\;\;Your\;\;\:Answers}}$

$\underline{\bold{Given}}$

$\bold{{x}^{2} + \frac{1}{ {x}^{2} }=51}$

$\bold{And\: We \:Need \:To\: Find...}$

$\bold{{x} - \frac{1}{x}- (Part-1)}$

And

$\bold{{x}^{3}-\frac{1}{{X}^{3}}-(Part-2)}$

So Now Let's Move For Solution ....

$\underline{\bold{Solution}}$

Now According To Question It's Given That ${x}^{2} + \frac{1}{ {X}^{2} } = 51$

Now To Find ...

$\bold{x - \frac{1}{x}}$

Follow The Simple Step ...

$\underline{\bold{Step-1)\:Find\:The\: Suitable\: Identity\:}}$

So The MosT Suitable Identity That Can Be Used Here Is...

$\boxed{\bold{{(a - b)}^{2} \:= {a}^{2} \:+ \:{b}^{2} -2ab}}$

$\underline{\bold{Step-2)\:Apply\: The \:Identity\: To \: Find\: Value \:Of\:\:{x}\:- \frac{1}{x} }}$

Now Applying This Identity ..

Where

$\underline{\bold{a\: = \:x \:\:And\: b \:= 1/x \:}}$

$\bold{ {(x\: - \:\frac{1}{x})}^{2} = \: {x}^{2} \:+ \:\frac{1}{ {x}^{2} } - 2(x) (\frac{1}{x} )}$

That Is ...

$\bold{{(x - \frac{1}{x})}^{2} = {x}^{2} + \frac{1}{ {x}^{2} } \: - 2 \: (by \: \: \: cancelling \: x)}$

So Now In Question Its Given That 

$\bold{ {x}^{2} + \frac{1}{ {x}^{2} }}$

Now Putting This Value In the Obtained Equation We Have ...

$\bold{ {(x - \frac{1}{x}) }^{2} = 51 - 2}$

That Is ...

$\bold{ {(x - \frac{1}{x} )}^{2} = 49}$

Now Moving Square To RHS We Have 

$\bold{x - \frac{1}{x} = \sqrt{49}}$

That Is ...

$\bold{x - \frac{1}{x} = 7}$

So Now We Have Value Of 

$\boxed{\bold{x - \frac{1}{x} = 7}}$

$\underline{\bold{Hence\:The\:Required\:Answer\:Is( Part-1)\:....}}$

$\boxed{\boxed{\bold{7}}}$

Now Let's Move For

$\underline{\bold{Part-2}}$

$\underline{\bold{Step -3)\:Solve\:For\:{x}^{3} -\frac {1}{{x}^{3}} }}$ 

Now By Identity ..

$\boxed{ \bold{{(a - b)}^{3} = {a}^{3} - {b}^{3} - 3ab \: (a - b)}}$

We Have ..

$\bold{ {(x - \frac{1}{x})}^{3} = {x}^{3} - \frac{1}{ {x}^{3} } - 3(x)( \frac{1}{x}) \;(x - \frac{1}{x})}$

That is ...

$\bold{{(x - \frac{1}{x} )}^{3} = {x}^{3} - \frac{1}{ {x}^{3} } - 3(x - \frac{1}{x} )}$

Now In Above Method We Have Found Value Of 

$\bold{x - \frac{1}{x} = 7}$

Now Putting This Value In This Obtained Equation We Have ...

$\bold{ {7}^{3} = {x}^{3} - \frac{1}{ {x}^{3} } - 3(7)}$

That Is ...

$\bold{343 = {x}^{3} - \frac{1}{ {x}^{3} } - 21}$

That is ...

$\bold{343 + 21 = {x}^{3} - \frac{1}{ {x}^{3} } }$

That Is ..

$\bold{364 = {x}^{3} - \frac{1}{ {x}^{3} } }$

Therefore Value Of 

$\boxed{\bold{ {x}^{3} - \frac{1}{ {x}^{3} } = 364}}$

$\bold{Hence\:The\: Required\:Answer\:of \: Second \:Part \: is ...}$

$\boxed{\mathfrak{\bold{364} } }$

$\Large{\bold{Thanks...}}$