If x = √3 -√2, then the value of x³ – 1/X³ is: (a) 22√3 (b) - 22√2 (c) 22√2 (d) -22√3
Answers
Option (b)
Step-by-step explanation:
Given :-
x = √3-√2
To find :-
Find the value of x³-(1/x³) ?
Solution :-
Given that
x = √3-√2 -------(1)
=> 1/x = 1/(√3-√2)
The Denominator = √3-√2
We know that
The Rationalising factor of √a-√b is √a+√b
The Rationalising factor of √3-√2 = √3+√2
On Rationalising the denominator then
=> 1/x = [1/(√3-√2)]×[(√3+√2)/(√3+√2)]
=> 1/x = (√3+√2)/(√3-√2)(√3+√2)
=> 1/x = (√3+√2)/[(√3)²-(√2)²]
Since , (a+b)(a-b) = a²-b²
Where, a = √3 and b = √2
=> 1/x = (√3+√2)/(3-2)
=> 1/x = (√3+√2)/1
=> 1/x = √3+√2 --------(2)
Now
The value of x³-(1/x³)
We know that
a³-b³ = (a-b)(a²+ab+b²)
=> x³-(1/x³) = [x-(1/x)](x²+(x/x)+(1/x²))
=> x³-(1/x³) = [x-(1/x)][x²+(1/x²)+1]
=> [(√3-√2)-(√3+√2)][(√3-√2)²+(√3+√2)²+1]
=> (√3-√2-√3-√2)(3+2-2√6+3+2+2√6+1)
=> (-√2-√2)(10+1)
=> (-2√2)(11)
=> -22√2
Answer:-
The value of x³-(1/x³) for the given problem is -22√2
Used Identities:-
→(a+b)² = a²+2ab+b²
→(a-b)² = a²-2ab+b²
→(a+b)(a-b) = a²-b²
→a³-b³ = (a-b)(a²+ab+b²)
Answer:
b)-22√2
Step-by-step explanation:
given,
x=√3-√2
therefore,
1/x=√3+√2
(x-1/x)=√3-√2-√3-√2
= -2√2
x^3-1/x^3 =(x-1/x)^3+3×x×1/x(x-1/x)
= -16√2-6√2
= -22√2