(2x+3y+4z) to the 3rd power
There is no easy rule, because there are many terms.
The following identity could help:
(a+b+c)^3
=c^3+3bc^2+3ac^2+3b^2c+6abc+3a^2c+b^3+3ab^2+3a^2b+a^3
So substitute 2x for a, 3y for b and 4z for c. Do not forget that a^2 would mean (2x)^2=4x^2, a^3 would mean (2x)^3=8a^3 and so on.
A good way to check your work is to add up the coefficients, they should equal (2+3+4)^3 = 729.
very patiently expand
(2x + 3y + 4z)(2x + 3y + 4z)(2x + 3y + 4z)
= ....
To raise the expression (2x + 3y + 4z) to the power of 3, you can use the binomial expansion or the distributive property. Here's the step-by-step process:
Step 1: Write down the expression to be raised to the power of 3:
(2x + 3y + 4z)
Step 2: Apply the binomial expansion or distributive property. For each term inside the parentheses, multiply it three times, combining like terms where applicable.
(2x + 3y + 4z) * (2x + 3y + 4z) * (2x + 3y + 4z)
Using the distributive property, we multiply each term from the first set of parentheses by each term from the second and third sets of parentheses:
(2x * 2x * 2x) + (2x * 2x * 3y) + (2x * 2x * 4z)
+ (2x * 3y * 2x) + (2x * 3y * 3y) + (2x * 3y * 4z)
+ (2x * 4z * 2x) + (2x * 4z * 3y) + (2x * 4z * 4z)
+ (3y * 2x * 2x) + (3y * 2x * 3y) + (3y * 2x * 4z)
+ (3y * 3y * 2x) + (3y * 3y * 3y) + (3y * 3y * 4z)
+ (3y * 4z * 2x) + (3y * 4z * 3y) + (3y * 4z * 4z)
+ (4z * 2x * 2x) + (4z * 2x * 3y) + (4z * 2x * 4z)
+ (4z * 3y * 2x) + (4z * 3y * 3y) + (4z * 3y * 4z)
+ (4z * 4z * 2x) + (4z * 4z * 3y) + (4z * 4z * 4z)
Step 3: Simplify each term by performing the multiplications and combining like terms.
8x^3 + 12x^2y + 16x^2z
+ 12xy^2 + 18xy^2 + 24xyz
+ 16xz^2 + 24xyz + 32z^3
+ 12x^2y + 18xy^2 + 24xyz
+ 18xy^2 + 27y^3 + 36yz^2
+ 24xyz + 36yz^2 + 48z^3
+ 16xz^2 + 24xyz + 32z^3
+ 24xyz + 36yz^2 + 48z^3
+ 32z^3
Therefore, the expanded form of (2x + 3y + 4z)^3 is:
8x^3 + 12x^2y + 16x^2z + 12xy^2 + 18xy^2 + 24xyz + 16xz^2 + 24xyz + 32z^3 + 12x^2y + 18xy^2 + 24xyz + 18xy^2 + 27y^3 + 36yz^2 + 24xyz + 36yz^2 + 48z^3 + 16xz^2 + 24xyz + 32z^3 + 24xyz + 36yz^2 + 48z^3 + 32z^3