Factor 27x^3-64y^3

Problem

27*x^3−64*y^3

Solution

1. Identify the expression as a difference of two cubes, which follows the general form a^3−b^3

2. Determine the cube roots of each term to find a and b

3. Calculate a by taking the cube root of 27*x^3 which results in a=3*x

4. Calculate b by taking the cube root of 64*y^3 which results in b=4*y

5. Apply the formula for the difference of cubes: a^3−b^3=(a−b)*(a^2+a*b+b^2)

6. Substitute the values of a and b into the formula.

7. Simplify the terms within the second factor: (3*x)^2=9*x^2 (3*x)*(4*y)=12*x*y and (4*y)^2=16*y^2

Final Answer

27*x^3−64*y^3=(3*x−4*y)*(9*x^2+12*x*y+16*y^2)

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