Please help me and do it in paper
If x=3+2√2 find the value of (√x-1/√x)^3
Answers
x = 3 + 2√2
1/x = 1/(3 + 2√2) = 1/(3 + 2√2) × (3 - 2√2)/(3 - 2√2)
1/x = (3 - 2√2)/(9 - 8)
1/x = 3 - 2√2
√x = √(3 + 2√2)
Now, we have to solve the term √(3 + 2√2). Here, neither you can multiply it by its rationalizing factor, for can you add or subtract anything, since it's enclosed in a square root.
For solving such terms, we need to split the enclosed term {here, it is 3 + 2√2} such that it is equal to the square of any digit.
3 + 2√2 = 1 + 2 + 2√2
It is similar to the identity:
(a + b)^2 = a^2 + b^2 + 2ab
Here, a = 1
B = √2
Thus:
3 + 2√2 = 1 + 2 + 2√2 = (1 + √2)^2
Usually, the identity used to split and compress the enclosed terms is the one used here. If not this one, the other one used is: (a - b)^2 = a^2 + b^2 - 2ab
The square root and square cancel each other. What is left is the value we need:
√x = (1 + √2) ___(1)
Now we've found √x. Similarly, we can find 1/√x
1/√x = √(1/x) = √(1/(3 + 2√2)) = √(3 - 2√2) = 1 - √2
Thus, 1/√x = (1 - √2) ____(2)
To find:
(√x - (1/√x))^3
Put (1) and (2):
=> ((1 + √2) - (1 - √2))^3
=> (1 + √2 - 1 + √2)^3
=> (√2)^3