Question

$\Large{\mathsf{Challenge:-}}$
$\large{\mathsf{a \sqrt{a} + b \sqrt{b} = 183 }}$
$\large{\mathsf{b \sqrt{a} + a \sqrt{b} = 182}}$
$\large{\mathsf{ \frac{9}{5}( a+b)}} =?$
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Answer 1

Step-by-step explanation:

Given :-

a√a+b√b =183

b√a+a√b = 182

To find :-

Find the value of 9/5(a+b)?

Solution :-

Given that :-

a√a+b√b =183 ------(1)

b√a+a√b = 182 -----(2)

Put √a = X and √b = Y then

=> a = X² and b = Y²

Now , equations becomes

(X²)(X)+(Y²)(Y) = 183

=> X³ + Y³ = 183 -----(3)

and

(Y²)(X)+(X²)(Y) = 182

=> XY²+X²Y = 182

=> XY(Y+X) = 182

=> XY(X+Y) = 182 -------(4)

Now

(3) we have

X³+Y³ = 183

We know that

(a+b)³ = a³+b³+3ab(a+b)

=> a³+b³ = (a+b)³-3ab(a+b)

=> (X+Y)³-3XY(X+Y) = 183

=> (X+Y)³-3(182) = 183 (from (4))

=> (X+Y)³ - 546 = 183

=> (X+Y)³ = 183+546

=> (X+Y)³ = 729

=> X+Y = ³√729

=> X+Y = 9 ------------(5)

Now, (4) becomes

=> XY (9) = 182

=> XY = 182/9 ------(6)

(5) can be written as

=> (√a+√b) = 9 ------(7)

Now

The value of (9/5)(√a+√b)

=> (9/5)(9)

=> (9×9)/5

=> 81/5

Answer:-

The value of (9/5)(√a+√b) for the given problem is 81/5

Used formulae:-

→ (a+b)³ = a³+b³+3ab(a+b)

→ a³+b³ = (a+b)³-3ab(a+b)