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Question

If the commutative property of multiplication is applied to the first term in the expression xy +xz and the distributive property is applied immediately afterwards, what is the result?

Answer 1

If we apply the commutative property to the first term in xy + xz, it becomes yx + xz. This is because the commutative property says that the order of the operands doesn't matter, so we reverse the order of x and y.

There is no real reason to apply the commutative property here if we are planning to apply the distributive. The common factor here is x; x is a factor of yx and a factor of xz. Pulling it out from yx leaves y and pulling it out from xz leaves z. The result is x(y + z). We would have gotten the same result if we hadn't applied the commutative property. If anything, applying the commutative made it harder to apply the distributive, because we put x into a different position in each term instead of leaving it as the first factor in both terms.

Answer 2

If you apply the comm. property to the first term you have yx + xz.
If the domain of the problem is the real numbers for x, y, and z there is no problem utilizing the distributive property, since the commutative property holds.
If this problem related to non-abelian groups (where the commutative property does not hold) then you would NOT be able to use the distributive property, which does happen to be independent of commutativity.
As a basic example, matrix multiplication is not commutative, so you would need to be careful doing what you describe above.

Answer 3

y * (xz) + x * (xz)

Answer 4

x(y+z)

Answer 5

x(y+z)