Question

suppose a,b are positive real number such that a√a+b√b=183,a√b+b√a=182. find 9/5 (a+b)

Answer 1

Answer:

Step-by-step explanation:

Given: a√a +b√b=183, a√b +b√a=182 or √(ab)(√a+√b)=182. Let’s assume u=√a and v=√b. The given equations are now free of annoying radicals viz.: u³+v³=183 ——-(a) & uv(u+v)=182 Or 3uv(u+v)=3*182=546 ——-(b). Add (a) & (b) to obtain, u³+v³+3uv(u+v)=(u+v)³=183+546=729=9³. We then claim that, u+v=9.

Going back to eqn (a), we’ve: u³+v³=183 or (u+v)(u²+v² -uv)=183 or (u+v)(u²+v²) -uv(u+v)=183 or (u+v)(u²+v²) - 182 = 183 or (u+v)(u²+v²) =365 or (u²+v²) = 365/(u+v)= 365/9. Then the required expression = (9/5)(a+b) = (9/5)(u²+v²)=(9/5)(365/9) = 73