if x^2-1/x=2, then find the value ofx^6-1/x^3
Answers
Answer:
Given,
x-1/x = 6 @1
The following calculation is necessary for later:
We know
(a+b)^2=(a-b)^2+4ab
Substituting a with x and b with 1/x
(x+1/x)^2=(x-1/x)^2+4.x.1/x
(x+1/x)^2=6^2+4
(x+1/x)^2=40
Therefore, (x+1/x)= Root of 40=6.325
My phone doesn't support typing root sign. Apologies!
By now we have values for both (x-1/x) and (x+1/x) as 6 and 6.32455 respectively.
The two equations we have in hand are:
1: x-1/x=6
2: x+1/x=6.32455
Adding both equations
2x=12.32455
Or x=6.1622775
Replacing value of x in x^6–1/x^6
We get 6.1622775^6–1/6.1622775^6
=54757.9914422–1/54757.9914422
=54757.9914422–0.00001826217
=54757.9
=54758 approx.
Through method of simplification:
x^6–1/x^6
Simplifying,
x^6 - 1/x^6
=(x^3)^2-(1/x^3)^2
Using a^2-b^2=(a+b)(a-b)
=(x^3-1/x^3)(x^3+1/x^3)
Using a^3-b^3=(a-b)^3+3ab(a-b) and a^3+b^3=(a+b)^3–3ab(a+b)
={(x-1/x)^3+3.x.1/x(x-1/x)}{(x+1/x)^3–3.x.1/x(x+1/x)}
={(x-1/x)^3+3(x-1/x)}{(x+1/x)^3–3(x+1/x)}
Substituting value for x-1/x for 6 and x+1/x for the root of 40
=(6^3+3*6)( 6.32455^3–3*6.32455) approx.
=(216+18)(252.98221–18.97366) approx.
=(234)*(234.00855) approx.
=54757.9 approx.
=54758
Step-by-step explanation: