Questions: Factor (64 m^3+27 v^3) completely.

Transcript text: Factor $64 m^{3}+27 v^{3}$ completely.

Solution

Solution Steps

To factor the expression $64m^3 + 27v^3$, recognize it as a sum of cubes. The sum of cubes formula is $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. Here, identify $a = 4m$ and $b = 3v$, then apply the formula to factor the expression.

Step 1: Identify the Expression

We start with the expression $64m^3 + 27v^3$. This expression is recognized as a sum of cubes.

Step 2: Apply the Sum of Cubes Formula

The sum of cubes can be factored using the formula: \[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \] In our case, we identify $a = 4m$ and $b = 3v$.

Step 3: Factor the Expression

Using the identified values of $a$ and $b$, we can factor the expression: \[ 64m^3 + 27v^3 = (4m + 3v)(16m^2 - 12mv + 9v^2) \]

Step 4: Verify the Factorization

To ensure the factorization is correct, we can expand the factored form: \[ (4m + 3v)(16m^2 - 12mv + 9v^2) = 64m^3 + 27v^3 \] This confirms that our factorization is accurate.

Final Answer

The complete factorization of the expression $64m^3 + 27v^3$ is: \[ \boxed{(4m + 3v)(16m^2 - 12mv + 9v^2)} \]

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