Question

estimate first , then name the sum or difference. use your estimate to check your work: 41+37 , 428 + 61 , 1314 + 272 , 1521 + 4325 , 33278 + 1902 , 9893 - 5164 , 27348 - 1267 , 1597 - 290 , 8001 - 457 , 55662 - 3465. "please answer them fully explained."

Answer 1

x+iy=a+ib/a-ib

or, x+iy=(a+ib)(a+ib)/(a-ib)(a+ib)

or, x+iy=(a²+2aib+i²b²)/(a²-i²b²)

or, x+iy={(a²-b²)+i.2ab}/(a²+b²)

or, x+iy=(a²-b²)/(a²+b²)+i. 2ab/(a²+b²)

∴, equating both sides, x=a²-b²/a²+b² and y=2ab/a²+b²

∴, x²+y²

= {(a²-b²)/(a²+b²)}²+{2ab/(a²+b²)}²

= {(a²-b²)²+4a²b²}/(a²+b²)²

= (a⁴-2a²b²+b⁴+4a²b²)/(a²+b²)²

= (a⁴+2a²b²+b⁴)/(a²+b²)²

= (a²+b²)²/(a²+b²)²

= 1