if x-1/x=8 fid the values of x^2+1/x^2) and (x^4+1/x^4)
Answers
EXPLANATION.
⇒ (x - 1/x) = 8.
As we know that,
Squaring on both sides of the equation, we get.
⇒ (x - 1/x)² = (8)².
⇒ (x)² + (1/x)² - 2(x)(1/x) = 64.
⇒ x² + 1/x² - 2 = 64.
⇒ x² + 1/x² = 64 + 2.
⇒ x² + 1/x² = 66.
Again squaring on both sides of the equation, we get.
⇒ (x² + 1/x²)² = (66)².
⇒ (x²)² (1/x²)² + 2(x²)(1/x²) = 4356.
⇒ x⁴ + 1/x⁴ + 2 = 4356.
⇒ x⁴ + 1/x⁴ = 4356 - 2.
⇒ x⁴ + 1/x⁴ = 4354.
Step-by-step explanation:
Given:-
x - 1/x = 8
To find out:-
Value of x^2 + 1/x^2 and x^4 + 1/x^4
Solution:-
We have,
x - 1/x = 8
On, squaring on both sides, we get
→ (x - 1/x)^2 = (8)^2
Now, applying algebraic Identity because our expression in the form of ;
(a - b)^2 = a^2 - 2ab + b^2
Where, we have to put in our expression a = x and b = 1/x ,we get
→ x^2 - 2(x)(1/x) + (1/x)^2 = (8)^2
→ x^2 - 2 + (1/x)^2 = 64
→ x^2 - 2 + 1/x^2 = 64
→ x^2 + 1/x^2 = 64 + 2
→ x^2 + 1/x^2 = 66
Now, again squaring on both sides, we get
(x^2 + 1/x^2)^2 = (66)^2
Now, applying algebraic Identity because our expression in the form of;
(a + b)^2 = a^2 + 2ab + b^2
Where, we have to put in our expression a = x and b = 1/x , we get
→ (x^2)^2 + 2(x^2)(1/x^2) + (1/x^2)^2 = (66)^2
→ x^4 + 2(x^2)(1/x^2) + 1/x^4 = 4356
→ x^4 + 2 + 1/x^4 = 4356
→ x^4 + 1/x^4 = 4356 - 2
→ x^4 + 1/x^4 = 4354
Answer:-
Hence, the value of x^2 + 1/x^2 is 66 and x^4 + 1/x^4 is 4354.
Used formulae:-
(a - b)^2 = a^2 - 2ab + b^2
(a + b)^2 = a^2 + 2ab + b^2