if x =2+√3 find the value of x^3-1/x^3
Answers
Step-by-step explanation:
Given:-
x = 2+√3
To find:-
Find the value of x^3 - (1/x^3) ?
Solution:-
Method-1:-
Given that
x = 2+√3 ------(1)
On cubing both sides then
x^3 = (2+√3)^3
We know that
(a+b)^3 = a^3+b^3+3a^2b+3ab^2
x^3 = 2^3+(√3)^3+3(2)^2(√3)+3(2)(√3)^2
=>x^3 = 8+3√3+12√3+18
=>x^3 = 26+15√3------(2)
1/x = 1/(2+√3)
Denominator = 2+√3
We know that
Rationalising factor of a+√b = a-√b
Rationalising factor of 2+√3 = 2-√3
On Rationalising the denominator then
=> [1/(2+√3)]×[(2-√3)/(2-√3)]
=> (2-√3)/[(2+√3)(2-√3)]
Denominator is in the form of (a+b)(a-b)
Where a = 2 and b = √3
(a+b)(a-b)=a^2-b^2
=> (2-√3)/[2^2-(√3)^2]
=> (2-√3)/(4-3)
=> (2-√3)/1
=> 2-√3
There fore 1/x = 2-√3-------(3)
On cubing both sides then
1/x^3 = (2-√3)^3
We know that
(a-b)^3 = a^3-b^3-3a^2b+3ab^2
1/x^3 = 2^3-(√3)^3-3(2)^2(√3)+3(2)(√3)^2
=>1/x^3 = 8-3√3-12√3+18
=>1/x^3 = 26-15√3-----(4)
Now the value of x^3 -(1/x^3)
=> (26+15√3)-(26-15√3)
=> 26+15√3-26+15√3
=>15√3+15√3
=> 30√3
Therefore, x^3-(1/x)^3 = 30√3
Method -2:-
Given that
x = 2+√3
1/x = 1/(2+√3)
Denominator = 2+√3
We know that
Rationalising factor of a+√b = a-√b
Rationalising factor of 2+√3 = 2-√3
On Rationalising the denominator then
=> [1/(2+√3)]×[(2-√3)/(2-√3)]
=> (2-√3)/[(2+√3)(2-√3)]
Denominator is in the form of (a+b)(a-b)
Where a = 2 and b = √3
(a+b)(a-b)=a^2-b^2
=> (2-√3)/[2^2-(√3)^2]
=> (2-√3)/(4-3)
=> (2-√3)/1
=> 2-√3
There fore 1/x = 2-√3
We know that
a^3-b^3 = (a-b)^3+3ab(a-b)
x^3-(1/x^3) = (x- 1/x)^3 +3(x)(1/x)(x- 1/x)
=> x^3-(1/x^3) = (x - 1/x)^3 + 3(x -1/x)
=>[(2+√3)-(2-√3)]^3+3[(2+√3)-(2-√3)]
=> (2+√3-2+√3)^3 +3(2+√3-2+√3)
=> (√3+√3)^3+3(√3+√3)
=> (2√3)^3+3(2√3)
=>8×3√3 +6√3
=> 24√3+6√3
=> 30√3
Therefore, x^3-(1/x)^3 = 30√3
Answer:-
The value of x^3 -(1/x^3) for the given problem is 30√3
Used formulae:-
- Rationalising factor of a+√b = a-√b
- (a+b)(a-b)=a^2-b^2
- (a-b)^3 = a^3-b^3-3a^2b+3ab^2
- a^3-b^3 = (a-b)^3+3ab(a-b)