%5E1%2F2-expand.jpeg "(x+y)^1/2 Expand")
Expanding (x + y)^(1/2)
The expression (x + y)^(1/2) represents the square root of (x + y). While we can't simplify it further into a simple expression involving only x and y, we can expand it using the binomial theorem.
Binomial Theorem
The binomial theorem provides a formula for expanding expressions of the form (a + b)^n, where n is a positive integer. It states:
(a + b)^n = a^n + (n choose 1)a^(n-1)b + (n choose 2)a^(n-2)b^2 + ... + (n choose n-1)ab^(n-1) + b^n
Where (n choose k) represents the binomial coefficient, calculated as:
(n choose k) = n! / (k! * (n-k)!)
Applying the Binomial Theorem to (x + y)^(1/2)
While the binomial theorem is typically used for positive integer exponents, it can be extended to fractional exponents using the concept of generalized binomial coefficients.
For (x + y)^(1/2), we have:
(x + y)^(1/2) = x^(1/2) + (1/2 choose 1)x^(-1/2)y + (1/2 choose 2)x^(-3/2)y^2 + ...
Where the generalized binomial coefficients are calculated using the gamma function:
(1/2 choose k) = Γ(3/2) / (Γ(k+1) * Γ(3/2-k))
Infinite Series Expansion
The expansion of (x + y)^(1/2) using the binomial theorem results in an infinite series. This means the expansion will have an infinite number of terms.
Note: This expansion is only valid for certain values of x and y, specifically when |x| > |y|.
Practical Implications
The infinite series expansion of (x + y)^(1/2) provides a way to approximate the square root of a sum, particularly when one term dominates the other. This can be useful in various mathematical and scientific applications.
Example
For instance, to approximate the square root of 1.01, we can write:
(1.01)^(1/2) = (1 + 0.01)^(1/2)
And then use the binomial expansion:
(1 + 0.01)^(1/2) ≈ 1 + (1/2)(0.01) - (1/8)(0.01)^2 + ...
This gives us a close approximation to the actual value of 1.01^(1/2).
Conclusion
Expanding (x + y)^(1/2) using the binomial theorem results in an infinite series representation. This provides a powerful tool for approximating the square root of a sum and has applications in various fields.