Question

a and b are two positive integers such that a√a+b√b = 183 and a√b+b√a = 182. Find 9(a+b)/5
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Answer 1

Solution :

Let (sq root a) =x

x^2 = a ............. (1)

and (sq root b) =y

y^2 = b .............. (2)

Subsitute (1) & (2) in equations

x^3 + y^3 = 183............ (3)

x^2y + y^2x = 182 ...... (4)

= xy (x + y) = 182............... (5)

As

(x+y)^3=x^3 + y^3 + 3(x^2y + y^2x) ...... (Identity)

Now put value (3) & (4) in identity

(x+y)^3 =183 + 3(182)

(x+y)^3 = 729

x+y = 9 ................... (6)

Now, put the value (6) x+y=9 into (5)

xy(x+y)=182

9xy = 182

xy = 182 / 9

Also, (x−y)^2 = (x+y)^2 − 4xy

= 9^2−4(182/9)

= 81 − 4(182/9)

= (729 - 728)/9

= 1 / 9

Therefore, x − y = - + 1 / 3

Since we found out

x + y = 9 & x − y = - + 1 / 3

By elimination method

CASE 1.

When x − y = + 1 / 3

Adding both the equations,

( x + y = 9) + ( x − y = + 1 / 3)

2x = 9 + 1 / 3

2x = (21 + 1) / 3

2x = 22 / 3

x = 22 / 6

x = 14 / 3

Therefore, y = 13 / 3

CASE 2.

When x − y = −1 / 3

Adding both the equations,

( x + y = 9) + ( x − y = - 1 / 3)

2x = 9 − 1 / 3

2x = 26 / 3

x = 13 / 3

Therefore, y = 14 / 3

Now find out 9 / 5 (a + b)

= 9 / 5 (x^2 + y^2)

= 9 / 5 [(13 / 3)^2 + (14 / 3)^2]

= (9 / 5) × (169 / 9) + (196 / 9)

= (9 / 5) x ( 365 / 9)

= (9 x 365) / (5 x 9)

= 73