Show that ( a1/3 - b1/3) ( a2/3 +a1/3 * b1/3 + b2/3 )?
To simplify the expression (a^(1/3) - b^(1/3))(a^(2/3) + a^(1/3)b^(1/3) + b^(2/3)), we can use the identity for the difference of cubes, which states that a^3 - b^3 = (a - b)(a^2 + ab + b^2). Let's break this down:
Let's consider the expression (a^(1/3) - b^(1/3))(a^(2/3) + a^(1/3)b^(1/3) + b^(2/3)).
Step 1: Recognize the pattern
The expression consists of two terms multiplied together. The first term is (a^(1/3) - b^(1/3)), and the second term is (a^(2/3) + a^(1/3)b^(1/3) + b^(2/3)).
Step 2: Apply the difference of cubes identity
We can see that the first term (a^(1/3) - b^(1/3)) is in the form a^3 - b^3, where a = a^(1/3) and b = b^(1/3). By using the difference of cubes identity, we can rewrite it as the product of two terms:
(a^(1/3) - b^(1/3)) = (a^(1/9) - b^(1/9))(a^(2/3) + a^(1/3)b^(1/3) + b^(2/3))
Step 3: Simplify the expression
Now we have:
(a^(1/3) - b^(1/3))(a^(2/3) + a^(1/3)b^(1/3) + b^(2/3)) = (a^(1/9) - b^(1/9))(a^(2/3) + a^(1/3)b^(1/3) + b^(2/3))
Therefore, we have shown that (a^(1/3) - b^(1/3))(a^(2/3) + a^(1/3)b^(1/3) + b^(2/3)) can be simplified as (a^(1/9) - b^(1/9))(a^(2/3) + a^(1/3)b^(1/3) + b^(2/3)).
I think you want to show that it is (a-b), no?
since (x^3 - y^3) = (x-y)(x^2+xy+y^2)
just substitute x=a^1/3 and y=b^1/3
Or, just multiply it out longhand.
Or, finish asking the question.