To factor the expression [tex]\(343 v^3 + 27 w^6\)[/tex], we first recognize that it can be expressed as a sum of cubes. Recall that the sum of cubes formula is given by:
[tex]\[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\][/tex]
Our goal is to rewrite the given expression in the form [tex]\(a^3 + b^3\)[/tex].
1. Identify the cube terms:
[tex]\[343 v^3 + 27 w^6\][/tex]
2. Notice that:
[tex]\[343 = 7^3 \quad \text{and} \quad 27 = 3^3\][/tex]
Therefore:
[tex]\[343 v^3 = (7v)^3 \quad \text{and} \quad 27 w^6 = (3w^2)^3\][/tex]
3. Now, we can rewrite the given expression as:
[tex]\[(7v)^3 + (3w^2)^3\][/tex]
4. Here, [tex]\(a\)[/tex] represents [tex]\(7v\)[/tex] and [tex]\(b\)[/tex] represents [tex]\(3w^2\)[/tex].
5. Apply the sum of cubes formula [tex]\(a^3 + b^3 = (a + b)\left(a^2 - ab + b^2\right)\)[/tex]:
[tex]\[(7v)^3 + (3w^2)^3 = (7v + 3w^2)\left((7v)^2 - (7v)(3w^2) + (3w^2)^2\right)\][/tex]
6. Now, expand and simplify each term inside the second parenthesis:
[tex]\[
\begin{align*}
(7v)^2 &= 49v^2 \\
(7v)(3w^2) &= 21vw^2 \\
(3w^2)^2 &= 9w^4
\end{align*}
\][/tex]
Therefore:
[tex]\[
(7v + 3w^2)\left(49v^2 - 21vw^2 + 9w^4\right)
\][/tex]
7. Put it all together to get the final factored form:
[tex]\[
343 v^3 + 27 w^6 = (7v + 3w^2)(49v^2 - 21vw^2 + 9w^4)
\][/tex]
Thus, the expression [tex]\(343 v^3 + 27 w^6\)[/tex] factors to:
[tex]\[
(7v + 3w^2)(49v^2 - 21vw^2 + 9w^4)
\][/tex]
