Question

If 4x+5y=10 and xy=12 then evaluate 64x3+125y3​

Answer 1

ANSWER:

• Value of the above expression is (-6200)

GIVEN:

• 4x+5y = 10 ....(i)
• xy = 12. ....(ii)

TO FIND:

• Value of 64x³+125y³.

SOLUTION:

=> 4x+5y = 10

Squaring both sides we get;

=> (4x+5y)² = (10)²

=> (4x)²+(5y)²+2(4x)(5y) = 100

=> 16x²+25y²+40xy = 100

Putting xy = 12 from (ii)

=> 16x²+25y² +40(12) = 100

=> 16x²+25y² = 100-480

=> 16x²+25y² = (-380)

= 64x³+125y³

= (4x)³+(5y)³

= (4x+5y)[(4x)²+(5y)²-(4x)(5y)]

= (4x+5y)(16x²+25y²-20xy)

Putting the values we get;

= 10[-380-20(12)]

= 10(-380-240)

= 10(-620)

= -6200

NOTE:

Some important formulas:

(a+b)² = a²+b²+2ab

(a-b)² = a²+b²-2ab

(a+b)(a-b) = a²-b²

(a+b)³ = a³+b³+3ab(a+b)

(a-b)³ = a³-b³-3ab(a-b)

a³+b³ = (a+b)(a²+b²-ab)

a³-b³ = (a-b)(a²+b²+ab)

(a+b)² = (a-b)²+4ab

(a-b)² = (a+b)²-4ab

Answer 2

GIVEN:

Equation - 1 : 4x + 5y = 10

Equation - 2: xy = 12

TO EVALUATE:

6x³ + 125y³

SOLUTION:

Take the first equation, 4x + 5y = 10

Now, cube on both sides.

==> (4x + 5y)³ = (10)³

==> (4x)³ + (5y)³ + 3(4x × 5y)(4x + 5y) = 1000

[ (a + b)³ = a³ + b³ + 3ab(a + b) ]

==> 64x³ + 125y³ + 3(20xy) (4x + 5y) = 1000

==> 64x³ + 125y³ + 3[ 20 × 12 ] [ 10 ] = 1000

[ From eq - (1) and (2) ]

==> 64x³ + 125y³ + 3 × 240 × 10 = 1000

==> 64x³ + 125y³ + 30 × 240 = 1000

==> 64x³ + 125y³ + 7200 = 1000

==> 64x³ + 125y³ = 1000 - 7200

==> 64x³ + 125y³ = -6200

Therefore, we get the required solution.