Question

a√a + b√b = 183
a√b + b√a = 182
Find 9/5(a+b)
Please show the steps also.

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