SOLUTION: If x+y=4,xy=2 THEN X^6+Y^6=?

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Question 1095680: If x+y=4,xy=2 THEN X^6+Y^6=?

Found 2 solutions by jim_thompson5910, ikleyn:
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Given:
x+y = 4
xy = 2

We'll call the equations shown above to be equation (1) and equation (2) (in that order)

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Multiply both sides of equation (2) by 2

xy = 2

2*xy = 2*2

2xy = 4

We'll refer to this as equation (3).

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Square both sides of equation (1)

x+y = 4

(x+y)^2 = 4^2

x^2+2xy+y^2 = 16

x^2+4+y^2 = 16 ....note how the "2xy" term has been replaced with 4. This is valid due to equation (3)

x^2+4+y^2-4 = 12-4 ... subtract 4 from both sides

x^2+y^2 = 12

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We will refer to x^2+y^2 = 12 as equation (4).

Cube both sides of equation (4) to get the following:

x^2+y^2 = 12

(x^2+y^2)^3 = 12^3

(x^2)^3+3(x^2)^2(y^2)+3(x^2)(y^2)^2+(y^2)^3 = 1728

x^6+3x^4y^2+3x^2y^4+y^6 = 1728

(x^6+y^6)+3x^4y^2+3x^2y^4 = 1728

(x^6+y^6)+3x^2y^2(x^2+y^2) = 1728

(x^6+y^6)+3(xy)^2(x^2+y^2) = 1728

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Now make substitutions. We'll use equation (2) and equation (4)

(x^6+y^6)+3(xy)^2(x^2+y^2) = 1728

(x^6+y^6)+3(2)^2(x^2+y^2) = 1728 ... Use equation (2) to make the first substitution. Replace xy with 2.

(x^6+y^6)+3(2)^2(12) = 1728 ... Use equation (4) to make the next substitution. Replace x^2+y^2 with 12

(x^6+y^6)+144 = 1728

(x^6+y^6)+144-144 = 1728-144 ... Subtract 144 from both sides

x^6+y^6 = 1584

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Final Answer: 1584

Answer by ikleyn(52776)   (Show Source):
You can put this solution on YOUR website!
.
This problem is solved in two steps:

Step 1. If x+y = 4 and xy = 2 then = = = 16 - 4 = 12. Step 2. Thus we proved that = 12. It implies further = = = . In the last expression, replace x^2*y^2 by = 4 and replace by 12. Then you can continue this chain of equalities in this way = = = = = . Thus = = 1584.

Answer.   . . . then = 1584.

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For similar solved problems, see the lessons
    - HOW TO evaluate expressions involving , and
    - Advanced lesson on evaluating expressions
in this site.

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    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic "Evaluation, substitution".

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Free of charge online textbook in ALGEBRA-I
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