1. ## Equation

a sqrt a +b sqrt b = 183 and
a sqrt b + b sqrt a = 182

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2. ## Re: Equation

Let $x = \sqrt{a}$ and $y = \sqrt{b}$

So $a\sqrt{a} + b\sqrt{b} = 183$ becomes $x^3 + y^3 = 183$, while
$a\sqrt{b} + b\sqrt{a} = 182$ becomes $x^2y + y^2x = 182$.

Use the cubic expansion
$(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3 = x^3 + y^3 + 3(x^2y + y^2x)$
and plug things in. Can you take it from here?

01

3. ## Re: Equation

Pls continue if you don't mind. Thanks in advance

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4. ## Re: Equation

Use the cubic expansion
$(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3 = x^3 + y^3 + 3(x^2y + y^2x)$
and plug things in.

$(x + y)^3$
$= x^3 + 3x^2y + 3xy^2 + y^3$
$= x^3 + y^3 + 3(x^2y + y^2x)$
$= 183 + 3(182)$
$= 729$

So
$(x + y)^3 = 729$, which means that $x + y = 9$. Solve this for y to get $y = 9 - x$.

Plug 9 - x in for y in
$x^3 + y^3 = 183$ to get
$x^3 + (9 - x)^3 = 183$.
Solve for x.

Go back to $y = 9 - x$, and plug in your x-value to get y. Finally, plug into
$x = \sqrt{a}$ and $y = \sqrt{b}$
to find a and b.

No more hints from me! You should be able to figure the rest out.

01