simultaneous equations with irrational variables

Question

Solve the simultaneous equations $a\sqrt a+b\sqrt b=183$ and $a\sqrt b+b\sqrt a=182$ I made an attempt in vain to equate the coefficients and eliminate

Answer 1

Hint. Take $x = \sqrt a$, $y = \sqrt b$ and note that $(x+y)^3 = x^3 + y^3 + 3(x^2y+xy^2)$.

Answer 2

We know the values for two symmetric polynomial expressions in $x=\sqrt a$ and $y=\sqrt b$. These can be expressed in the elemntary symmetric polynomials$s=x+y$ and $p=xy$. If we have $s$ and $p$, we find $x,y$ as the roots of $X^2-pX+s$.

Now $$\begin{align}a\sqrt a+b\sqrt b&=x^3+y^3\\ &=(x+y)^3-3xy(x+y)\\&=s^3-3sp\end{align}$$ and $$\begin{align}a\sqrt b+b\sqrt a&=x^2y+y^2x\\ &=xy(x+y)\\&=ps\end{align}$$ We conclude that $s=\sqrt[3]{183+3\cdot 182}$ and then $p=\frac{182}s$.

Answer 3

-Take eq 1; -square both sides; -take eq 2; -square both sides; -use the formula (a+b)^3 = a^3+ b^3 +3(a^2 b+a b^2) -once you have equated that you get; (a+b)^3 = 132861 -taking 3root. you get a+b = 51.