Question

a√a+b√b =183 a√b+b√a =182 find 9÷5(a+b)

Answer 1

Hey Mate !

Here is your solution :

Given,

=> a√a + b√b = 182 ----------- ( 1 )
________________________

=> a√a + b√b = 183 -------- ( 2 )

=> (√a )³ + (√b )³ = 183
_________________________

Using identity :

=> ( x³ + y³ ) = ( x + y )³ - 3xy( x + y )
__________________________

=> (√a +√b )³ - 3 √a ×√b (√a + √b ) = 183

=> (√a +√b )³ - 3a√b - 3b√a = 183 -----( 3 )

Taking out ( -3 ) as common ,

=> (√a +√b )³ - 3( a√b + b√a ) = 183

=> (√a +√b )³ - 3 × 182 = 183

=> (√a +√b )³ - 546 = 183

=> (√a + √b )³ = 183 + 546

=> (√a + √b )³ = 729

=> (√a + √b )³ = ( 9 )³

As exponent is equal , so base must be equal,

•°• (√a + √b ) = 9 -------- ( 4 )

Squaring both sides ,

=> ( √a + √b )² = ( 9 )²
__________________________

Using identity :

=> ( a + b )² = a² + b² + 2ab
__________________________

=> (√a )² + (√b )² + 2 √a × √b = 81

=> a + b + 2√ab = 81 ---------- ( 5 )

Continuing the ( 3 ),

=> ( √a + √b )³ - 3a√b - 3b√a = 183

Taking out { -3√(ab) } as common,

=> ( √a + √b )³-{ 3√( ab )} (√a + √b ) = 183

Substituting the value of ( 4 ),

=> ( 9 )³ - { 3√( ab ) } ( 9 ) = 183

=> 729 - 27√( ab ) = 183

=> 729 - 183 = 27√( ab )

=> 546 = 27√( ab )

=> 546 ÷ 27 = √( ab )

=> 182 / 9 = √( ab ) -------- ( 6 )

By substituting the value of ( 6 ) in ( 5 ),

=> a + b + 2√( ab ) = 81

=> ( a + b ) + 2 ( 182 / 9 ) = 81

=> ( a + b ) + ( 364 / 9 ) = 81

=> ( a + b ) = 81 - ( 364 / 9 )

=> ( a + b ) = ( 729 - 364 ) / 9

=> ( a + b ) = ( 365 / 9 ) ---------- ( 7 )

Now,

= ( 9/5 ) ( a + b )

By putting the value of ( 7 ),

= ( 9/5 ) ( 365 / 9 )

= [ ( 365 × 9 ) / ( 5 × 9 ) ]

= 73

Hence, the required answer is 73.

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Hope it helps !! ^_^