Question

if x^2+1/81x^2= 2, find 3x-1/3x and 81x^4+1/81x^4



Correct answer will be marked brainiest.......​

Answer 1

9x2 + (1/9x2) = (3x – (1/3x))2 + 2

= (5)2 + 2 = 25 + 2 = 27

(ii) 81x4 + (1/81x4) = (9x2 + (1/9x2))2 – 2

= (27)2 – 2 = 729 – 2 = 727

Answer 2

Answer:

$3x-\frac{1}{3x} = 4$    and   $81x^{4}+\frac{1}{81x^{4}} = 322$

Explanation:

Ok here's your solution....

I mean well defined solution......

So let's consider $x^{2}+\frac{1}{81x^{2}}=2$ as equation 1

After that,

Let's make a equation according to the identity

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

So,

Let's take

a = 3x and,

b =  $\frac{1}{3x}$

Then according to the identity,

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

3x in the last of the eq gets cancelled so final equation,

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

SOLVING R.H.S

⇒$(9x)^{2}+\frac{1}{9x^{2}} -2$

Now after this we have the value of 2 as

$x^{2}+\frac{1}{81x^{2}}$                          (from equation 1)

So,

$(9x)^{2}+\frac{1}{9x^{2}} -[x^{2}+\frac{1}{81x^{2}}]$

⇒$9x^{2} - x^ {2}+\frac{1}{9x^{2}} - \frac{1}{81x^{2}}$

⇒$8x^{2}+\frac{9}{81x^{2}} - \frac{1}{81x^{2}}$

⇒$8x^{2}+\frac{8}{81x^{2}}$

Taking 8 as common,

⇒$8(x^{2}+\frac{1}{81x^{2}} )$

We have,

$x^{2}+\frac{1}{81x^{2}} = 2$

∴$8(x^{2}+\frac{1}{81x^{2}} )$  ⇒ $8(2)$   ⇒  $16$

So we get 16 in the R.H.S

And we have $(3x-\frac{1}{3x}) ^{2}$ in the L.H.S

To get $3x-\frac{1}{3x}$,

⇒$3x-\frac{1}{3x} = \sqrt{16}$

⇒$3x-\frac{1}{3x} = 4$

So answer of the first one is 4

Now squaring both sides we get,

⇒$9x^{2}+\frac{1}{9x^{2}}-2(9x^{2})(\frac{1}{9x^{2}} ) = 16$

⇒$9x^{2}+\frac{1}{9x^{2}}-2 = 16$

⇒$9x^{2}+\frac{1}{9x^{2}}= 16+2$

⇒$9x^{2}+\frac{1}{9x^{2}} = 18$

Again squaring both sides,

$(9x^{2}+\frac{1}{9x^{2}})^{2} = (18)^{2}$

⇒$81x^{4}+\frac{1}{81x^{4}} + 2(81x^{4})(\frac{1}{81x^{4}}) = 324$

⇒$81x^{4}+\frac{1}{81x^{4}} + 2 = 324$

⇒$81x^{4}+\frac{1}{81x^{4}} = 324 - 2$

⇒$81x^{4}+\frac{1}{81x^{4}} = 322$

So the answer of 2nd one is 322.

Hope this helps....

Please mark as brainliest the answer is 100% correct