To factor the expression \( 27x^3 - 64y^3 \), we recognize that it is a difference of cubes. The difference of cubes can be factored using the formula:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
In this case, we have:
- \( a = 3x \) (since \( (3x)^3 = 27x^3 \))
- \( b = 4y \) (since \( (4y)^3 = 64y^3 \))
Applying the formula:
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First, we calculate \( a - b \): \[ 3x - 4y \]
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Next, we find \( a^2 + ab + b^2 \):
- \( a^2 = (3x)^2 = 9x^2 \)
- \( ab = (3x)(4y) = 12xy \)
- \( b^2 = (4y)^2 = 16y^2 \)
So, \[ a^2 + ab + b^2 = 9x^2 + 12xy + 16y^2 \]
Thus, the factored form of \( 27x^3 - 64y^3 \) is:
\[ (3x - 4y)(9x^2 + 12xy + 16y^2) \]
The linear factor of the expression is therefore:
\[ 3x - 4y \]
So the response is:
3 x minus 4 y.
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