Question

X^2+y^2=2 x^6+y^6+6x^2y^2=?

Answer 1

Ответ:

8

Пошаговое объяснение:

x⁶+y⁶+6x²y²=(x²)³+(y²)³+6x²y²=(x²+y²)(x⁴+y⁴-x²y²)+6x²y² =

взамен х²+ у² подставляем значение 2.

=2*((x²+y²)²-3x²y²)+6x²y²= аналогично

=2*(2²-3x²y²)-6x²y²=2³-6x²y²+6x²y²=2³=8

а³+b³=(a+b)(a²+b²-ab)

(а+b)²=a²+b²+2ab

Answer 2

To find the value of $x^6 + y^6 + 6x^2y^2$, we need to express it in terms of the equation $x^2 + y^2 = 2$. Let's simplify the expression step by step.

We know that $(x^2 + y^2)^3 = (x^2 + y^2)(x^2 + y^2)^2 = 2(x^2 + y^2)^2$.

Expanding both sides of the equation, we get:

$(x^2 + y^2)^3 = (x^2 + y^2)^2 \cdot (x^2 + y^2)$

$(x^2 + y^2)(x^2 + y^2)(x^2 + y^2) = 2(x^2 + y^2)^2$

$x^6 + 2x^4y^2 + 2x^2y^4 + y^6 = 2(x^4 + 2x^2y^2 + y^4)$

Now, let's substitute $x^2 + y^2$ with 2 (from the given equation):

$x^6 + 2x^4y^2 + 2x^2y^4 + y^6 = 2(x^4 + 2x^2y^2 + y^4)$

We can simplify further by canceling out the common terms:

$x^6 + y^6 + 6x^2y^2 = 2x^4 + 4x^2y^2 + 2y^4$

$x^6 + y^6 + 6x^2y^2 = 2(x^4 + 2x^2y^2 + y^4)$

Now, substituting $x^2 + y^2 = 2$ once more:

$x^6 + y^6 + 6x^2y^2 = 2(2^2)$

$x^6 + y^6 + 6x^2y^2 = 2 \cdot 4$

$x^6 + y^6 + 6x^2y^2 = 8$

Therefore, $x^6 + y^6 + 6x^2y^2 = 8$ when $x^2+y^2=2$ 0 0