is (a+b)^3 equal to a^3+b^3?

Nope.

Try it yourself with a=1, b = 2
(1+2)^3 = 3^3 = 27
1^3 + 2^3 = 1 + 8 = 9

or

(1-2)^3 = (-1)^3 = -1
1^3 - 2^3 = 1 - 8 = -7

To multiply out (a+b)^3 ...
(a+b)(a+b)(a+b) =
(a+b)(a^2 + ab + ab + b^2) = (a+b)(a^ + 2ab + b^2) =
a^3 + 2a^2b + ab^2 + a^2b + 2ab^2 + b^3 =
a^3 + 3a^2b + 3ab^2 + b^3 =
a^3 + b^3 + 3a^2b + 3ab^2

To multiply out (a-b)^3
(a-b)(a-b)(a-b)
(a-b)(a^2 - ab - ab + b^2) = (a-b)(a^2 - 2ab + b^2)
a^3 - 2a^2b + ab^2 - a^2b + 2ab^2 - b^3 =
a^3 - 3a^2b + 3ab^2 - b^3 =
a^3 - b^3 - 3a^2b + 3ab^2

False - you can test this by putting in numbers and seeing what happens:

(a + b)³ = a³ + b³

I'll use 2 for a and 3 for b:

(2 + 3)³ = 2³ + 3³
5³ = 8 + 27
125 = 35

FALSE


The link I pasted below shows the proper way to factor out sums and differences of cubes:

a³ + b³ = (a + b)(a² - ab + b²)
a³ - b³ = (a - b)(a² + ab + b²)

(a + b)^3
= (a + b)(a + b)(a + b)
= (a*a + b*a + a*b + b*b)(a + b) (* = multiply)
= (a^2 + ab + ab + b^2)(a + b)
= (a^2 + 2ab + b^2)(a + b)
= a^2*a + 2ab*a + b^2*a + a^2*b + 2ab*b + b^2*b
= a^3 + 2a^2b + ab^2 + a^2b + 2ab^2 + b^3
= a^3 + 2a^2b + a^2b + ab^2 + 2ab^2 + b^3
= a^3 + 3a^2b + 3ab^2 + b^3

yes, I'm pretty sure
because when you distribute the ^3 in the first part of the equation, you come out with a^3+b^3

so I think it's equal

No !
(a + b)³

(a + b)(a² + 2ab + b²)

a³ + 2a²b + ab²
____a²b + 2ab² + b³

a³ + 3a²b + 3ab² + b³

Similarly
(a - b)³

(a - b)(a² - 2ab + b²)

a³ - 2a²b + ab²
__ -a²b + 2ab² - b³

a³ - 3a²b + 3ab² - b³

They are equal when b and a are both zero, otherwise they are seldom equal. For example if a=1 and b=1 the left hand side is (1+1)^3 = 2^3 = 8, but the right side is 1^3 + 1^3 = 2.

That is how you can answer questions like this yourself, just plug in some easy numbers. If both sides have the same value it might be true in general, but it is not a proof. If the two sides have different values, then for sure it is false.

no
(a+b)^3 = a^3+3a^2b+3ab^2+b^3
and (a-b)^3= a^3-3a^2b+3ab^2-b^3

no

(a+b)^3 = ((a+b)*(a+b)*(a+b) = 1a^3+ 3a^2b 3ab^2 + 1b^3

(a-b)^3=(a-b)*(a-b)*(a-b)

i'll do this slowly
((a-b)*(a-b) = a^2 -2ab + b^2

a^2 -2ab + b^2*(a-b)

= a^3 -2a^2b +ab^2 -a^2b -2aB^2 -b^3
= a^3 -3a^2b -1ab^2 -b^3

No,
(a+b)^3=a^3+3a^2b+3ab^2+b^3
(a-b)^3=a^3-3a^2b+3ab^2-b^3
Multiply (a+b)(a+b) to verify this.
a^3+b^3=(a+b)(a^2-ab+b^2)
a^3-b^3=(a-b)(a^2-ab+b^2)