If [x^2 +(1/x^2)=7] what is the value of [7x^3+8x-(7/x^3)-(8/x)]I know the answer which is ±64 But I want a step-by-step explaination on this...

We'll factorize by 7 the terms 7x^3 and
7/x^3:

7(x^3 - 1/x^3)

We'll
factorize by 8 the terms 8x and 8/x:

8(x -
1/x)

We'll apply the
identity:

a^3 - b^3 = (a-b)(a^2 + ab +
b^2)

(x^3 - 1/x^3) = (x - 1/x)(x^2 + x/x +
1/x^2)

But, form enunciation, x^2 + 1/x^2 =
7.

(x^3 - 1/x^3) = (x - 1/x)(7 +
1)

(x^3 - 1/x^3) = 8(x -
1/x)

7*(x^3 - 1/x^3) = 7*8(x -
1/x)

The expression to be calculated will
become:

7(x^3 - 1/x^3) + 8(x - 1/x) = 7*8(x - 1/x) + 8(x -
1/x)

We'll factorize by 8(x -
1/x):

7(x^3 - 1/x^3) + 8(x - 1/x) = 8(x -
1/x)*(7+1)

7(x^3 - 1/x^3) + 8(x - 1/x) = 64(x -
1/x)

We'll calculate x -
1/x:

We'll raise to square x -
1/x:

(x - 1/x)^2 = x^2 - 2x/x +
1/x^2

But x^2 + 1/x^2 = 7.

(x
- 1/x)^2 = 7 - 2

(x - 1/x)^2 = 5 => x - 1/x = sqrt
5

The value of expression will
be:

7(x^3 - 1/x^3) + 8(x - 1/x) = 64*sqrt
5

The value of the given expression
is:7x^3+8x-(7/x^3)-(8/x) = 64*sqrt 5.