Math  /  Algebra

Question姣, All factors in your answer should have integer coefficients. 512w3+27x3=512 w^{3}+27 x^{3}= \square

Studdy Solution

STEP 1

1. The expression 512w3+27x3512 w^3 + 27 x^3 is a sum of cubes.
2. We will use the sum of cubes formula to factor the expression.
3. The sum of cubes formula is: a3+b3=(a+b)(a2ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2).

STEP 2

1. Identify aa and bb in the expression 512w3+27x3512 w^3 + 27 x^3.
2. Apply the sum of cubes formula to factor the expression.

STEP 3

Identify aa and bb such that a3=512w3a^3 = 512 w^3 and b3=27x3b^3 = 27 x^3.
a=512w33=8w a = \sqrt[3]{512 w^3} = 8w b=27x33=3x b = \sqrt[3]{27 x^3} = 3x

STEP 4

Apply the sum of cubes formula a3+b3=(a+b)(a2ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2).
512w3+27x3=(8w+3x)((8w)2(8w)(3x)+(3x)2) 512 w^3 + 27 x^3 = (8w + 3x)((8w)^2 - (8w)(3x) + (3x)^2)

STEP 5

Calculate each part of the factored expression:
(8w)2=64w2 (8w)^2 = 64w^2 (8w)(3x)=24wx (8w)(3x) = 24wx (3x)2=9x2 (3x)^2 = 9x^2
So, the expression becomes:
(8w+3x)(64w224wx+9x2) (8w + 3x)(64w^2 - 24wx + 9x^2)
The factored expression is:
(8w+3x)(64w224wx+9x2) \boxed{(8w + 3x)(64w^2 - 24wx + 9x^2)}

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