Question

use the binomial theorem to expand the binomial:

\\(-x^{-1} + 2y^{-1})^4\\

Explanation:

Step1: Apply the Binomial Theorem

$$(-x^{-1} + 2y^{-1})^4 = \sum_{k=0}^{4} \binom{4}{k} (-x^{-1})^{4-k} (2y^{-1})^k$$

Step2: Expand each term

$$\binom{4}{0}(-x^{-1})^4(2y^{-1})^0 + \binom{4}{1}(-x^{-1})^3(2y^{-1})^1 + \binom{4}{2}(-x^{-1})^2(2y^{-1})^2 + \binom{4}{3}(-x^{-1})^1(2y^{-1})^3 + \binom{4}{4}(-x^{-1})^0(2y^{-1})^4$$

Step3: Simplify the coefficients

$$1(x^{-4})(1) + 4(-x^{-3})(2y^{-1}) + 6(x^{-2})(4y^{-2}) + 4(-x^{-1})(8y^{-3}) + 1(1)(16y^{-4})$$

Step4: Write final expression

$$x^{-4} - 8x^{-3}y^{-1} + 24x^{-2}y^{-2} - 32x^{-1}y^{-3} + 16y^{-4}$$

Answer:

\(x^{-4} - 8x^{-3}y^{-1} + 24x^{-2}y^{-2} - 32x^{-1}y^{-3} + 16y^{-4}\)