if x²+1/x² = 18, find x³ - 1/x³
Answers
Answer:
76
Step-by-step explanation:
Step-by-step explanation:
Given:
{x}^{2} + \dfrac{1}{ {x}^{2} } = 18x
2
+
x
2
1
=18
To find:
{x}^{3} - \dfrac{1}{ {x}^{3} }x
3
−
x
3
1
Solution:
{x}^{2} + \dfrac{1}{ {x}^{2} } = 18x
2
+
x
2
1
=18
[subtracting 2 from both side
x
2
+
x
2
1
−2=18−2
x
2
−2.x.
x
1
+
x
2
1
=16
Hence, The required answer is 76.
Step-by-step explanation:
Given:-
x^2+1/x^2 = 18
To find:-
find the value of x^3 - 1/x^3?
Solution:-
Given that
x^2 +1/x^2 = 18 -----------(1)
we know that
(a-b)^2 = a^2-2ab+b^2
=>a^2+b^2 = (a-b)^2+2ab
Where , a= x^2 and b = 1/x^2
=>x^2 + (1/x^2) = [x-(1/x)]^2 +2(x)(1/x)
=>18 = [x - (1/x)]^2 +2
=>[x-(1/x)]^2 = 18-2
=>[x-(1/x)]^2 = 16
=>x-(1/x) = √16
=>x-(1/x)=4 ------------(2)
(on taking positive value )
Now
The value of x^3-(1/x^3)
We know that
a^3 - b^3 = (a-b)(a^2 +ab +b^2)
Where , a= x and b = 1/x
=>x^3 - (1/x)^3 = [x-(1/x)][x^2+(x)(1/x)+(1/x)^2]
=>x^3 - (1/x)^3 = [x-(1/x)][x^2+1+(1/x)^2]
=>x^3 - (1/x)^3 = [x-(1/x)][x^2+(1/x)^2 +1]
=>x^3 - (1/x)^3 = (4)(18+1)
=>x^3 - (1/x)^3 = 4(19)
=>x^3 - (1/x)^3 = 76
Answer:-
The value of x^3 - (1/x)^3 for the given problem is
76
Used formulae:-
- (a-b)^2 = a^2-2ab+b^2
- a^2+b^2 = (a-b)^2+2ab
- a^3 - b^3 = (a-b)(a^2 +ab +b^2)