Question

(a2+b2)(a4-a2b2+b4)-(a3-b3)(a3+b3)=2b6

Answer 1

$\underbrace {(a^2+b^2)(a^4-a^2b^2+b^4)}_{(a^2)^3+(b^2)^3}-\underbrace {(a^3-b^3)(a^3+b^3)}_{(a^3)^2-(b^3)^2}=\\\\=(a^2)^3+(b^2)^3-\Big ((a^3)^2-(b^3)^2\Big )=a^6+b^6-a^6+b^6=2b^6\\\\\\\star \; \; (x+y)^3=(x+y)(x^2-xy+y^2)\; \; \star \\\\\star \; \; (x-y)(x+y)=x^2-y^2\; \; \star$

Answer 2

(a²+b²)(a⁴-a²b²+b⁴)-(a³-b³)(a³+b³)=

(a²)³+(b²)³-(a³)²+(b³)²=

a^6+b^6-a^6+b^6=2b^6

Answer 3

Let's verify the given equation step by step:

1. (a^2 + b^2)(a^4 - a^2b^2 + b^4) - (a^3 - b^3)(a^3 + b^3)

Now, let's expand each term:

1. (a^2 + b^2)(a^4 - a^2b^2 + b^4) = a^6 - a^4b^2 + a^2b^4 + a^2b^2 - ab^4 + b^6

2. (a^3 - b^3)(a^3 + b^3) = a^6 + a^3b^3 - a^3b^3 - b^6

Now, let's simplify the equation:

(a^2 + b^2)(a^4 - a^2b^2 + b^4) - (a^3 - b^3)(a^3 + b^3) = (a^6 - a^4b^2 + a^2b^4 + a^2b^2 - ab^4 + b^6) - (a^6 - b^6)

Now, some terms cancel out:

= a^6 - a^4b^2 + a^2b^4 + a^2b^2 - ab^4 + b^6 - a^6 + b^6

The a^6 and -a^6 terms cancel each other out, and the b^6 and -b^6 terms cancel each other out as well:

= - a^4b^2 + a^2b^4 + a^2b^2 - ab^4

Next, factor out the common term 'ab^2':

= ab^2(-a^3 + ab^2 + a - b^3)

Now, we can see that the expression inside the parentheses is equivalent to zero:

• a^3 + ab^2 + a - b^3 = 0

Therefore, the entire expression simplifies to zero:

ab^2(-a^3 + ab^2 + a - b^3) = 0

The given equation is not equal to 2b^6; it simplifies to zero.