a-a-b-b-183-and-b-a-a-b-182-What-is-the-value-of-9-5-a-b-

Question Number 191486 by MATHEMATICSAM last updated on 24/Apr/23

a \sqrt{a} + b \sqrt{b} = 183 and b \sqrt{a} + a \sqrt{b} = 182

What is the value of \frac{9}{5} (a + b) ?

Answered by mr W last updated on 24/Apr/23

l e t A = \sqrt{a}

B = \sqrt{b}

{A B}^{2} + A^{2} B = 182

\Rightarrow A B (A + B) = 182

A^{3} + B^{3} = 183

{(A + B)}^{3} - 3 A B (A + B) = 183

{(A + B)}^{3} - 3 \times 182 = 183

\Rightarrow A + B = \sqrt[3]{183 + 3 \times 182} = 9

a + b = A^{2} + B^{2} = {(A + B)}^{2} - 2 A B

= 9^{2} - 2 \times \frac{182}{9} = \frac{365}{9}

\frac{9}{5} (a + b) = \frac{9}{5} \times \frac{365}{9} = 73

Commented by mehdee42 last updated on 24/Apr/23

t h a t w a s p e r f e c t