Question 1141270
<pre>System 1:     {{{system(a*sqrt(a) + b*sqrt(b) = 183, 
 a*sqrt(b) + b*sqrt(a) = 182)}}} 

Let {{{x=sqrt(a)}}}, {{{x^2=a}}},  {{{y=sqrt(b)}}}, {{{y^2=b}}}, then

{{{system(x^2*x + y^2*y = 183, 
 x^2*y + y^2*x = 182)}}}

System 2:     {{{system(x^3 + y^3 = 183, 
 x^2*y + x*y^2 = 182)}}}

Multiply the 2nd equation by 3

{{{system(x^2*x + y^2*y = 183, 
 3x^2*y + 3y^2*x = 546)}}}

Add the two equations:

{{{x^3+3x^2*y+3x*y^2+y^3=729}}}

Factor the left side:
 
{{{(x+y)^3=729}}}

Take cube roots of both sides:

Eq. 1     {{{x+y=9}}}

Factor the left side of the 1st equation in system 2

{{{x^3 + y^3 = 183}}}
{{{(x+y)(x^2-xy+y^2)=183}}}

Use Eq. 1 to substitute 9 for (x+y)

{{{9(x^2-xy+y^2)=183}}}
{{{x^2-xy+y^2=183/9}}}, reduce 183/9 to 61/3

Eq. 2     {{{x^2-xy+y^2=61/3}}}

Square both sides of Eq. 1:

{{{(x+y)^2=9}}}

Eq. 3     {{{x^2+2xy+y^2=81}}}

Multiply Eq. 2 by -1 and add Eq. 3

{{{""-x^2+xy-y^2=-61/3}}}
 {{{x^2+2xy+y^2=81}}}

{{{3xy=-61/3+81}}}
{{{3xy=-61/3+243/3}}}
{{{3xy=182/3}}}
{{{xy=182/9}}}

Substitute {{{182/9}}} for xy in in Eq. 2

{{{x^2-182/9+y^2=183/9}}}

{{{x^2+y^2=183/9+182/9}}}

{{{x^2+y^2=365/9}}}

Since x²=a and y²=b,

{{{a+b=365/9}}}

{{{expr(9/5)(a+b)=(9/5)(365/9) = 73}}}

Edwin</pre>