An algebra problem by swastik p (2)

Algebra Level pending

Suppose a , b ∈ R + a,b\in\mathbb{R^+} such that a a + b b = 183 and a b + b a = 182 a\sqrt{a}+b\sqrt{b}=183~~~~\text{and}~~~~ a\sqrt{b}+b\sqrt{a}=182 .

What is the value of 9 5 ( a + b ) ? \dfrac{9}{5}\left(a+b\right)~~~?

65 50 73 34 77

S P by s p , 16, India

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1 solution

X X
May 27, 2018

Let a = x , b = y \sqrt{a}=x,\sqrt{b}=y ,then x 3 + y 3 = 183 , x 2 y + y 2 x = 182 x^3+y^3=183,x^2y+y^2x=182

x 3 + y 3 + 3 ( x 2 y + y 2 x ) = ( x + y ) 3 = 183 + 182 × 3 = 729 , x + y = 9 x^3+y^3+3(x^2y+y^2x)=(x+y)^3=183+182\times3=729,x+y=9

x 3 + y 3 + x 2 y + y 2 x = ( x 2 + y 2 ) ( x + y ) = ( a + b ) × 9 = 183 + 182 = 365 , a + b = 365 9 x^3+y^3+x^2y+y^2x=(x^2+y^2)(x+y)=(a+b)\times9=183+182=365,a+b=\frac{365}9

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