Question

If x^3+ y^3 = 9 and x + y = 3, then the value of x^4+y^4 is?

Answer 1

Answer:

x⁴ + y⁴ = 17

Step-by-step explanation:

Given ---> x³ + y³ = 9 and x + y = 3

To find ---> Value of ( x⁴ + y⁴ )

Solution---> We know that

x³ + y³ = ( x + y ) ( x² + y² - xy )

Putting x³ + y³ = 9 and x + y = 3 we get

=> 9 = 3 ( x² + y² - xy )

=> x² + y² - xy = 9 / 3

=> x² + y² - xy = 6 .................... (1)

Adding and subtracting 2xy we get

=> ( x² + y² + 2 xy ) - 2 xy - xy = 3

We have an identity

a² + b² + 2 ab = ( a + b )² , applying it here

=> ( x + y )² - 3 xy = 3

=> ( 3 )² - 3 xy = 3

=> 9 - 3 xy = 3

=> 3 xy = 9 - 3

=> xy = 6 / 3

=> xy = 2

By relation (1)

x² + y² - xy = 3

=> x² + y² - 2 = 3

=> x² + y² = 3 + 2

=> x² + y² = 5

Squaring both sides we get

=> ( x² + y² )² = (5)²

Applying , (a + b )² = a² + b² + 2ab , here we get

=> ( x² )² + ( y² )² + 2 x² y² = 25

=> x⁴ + y⁴ + 2 x² y² = 25

=> x⁴ + y⁴ + 2 ( x y )² = 25

Putting xy = 2 we get

=> x⁴ + y⁴ + 2 ( 2 )² = 25

=> x⁴ + y⁴ + 2 ( 4 ) =25

=> x⁴ + y⁴ + 8 = 25

=> x⁴ + y⁴ = 25 - 8

=> x⁴ + y⁴ = 17