To factor the expression [tex]\(x^3 y^3 + 343\)[/tex] using the sum of cubes formula, follow these steps:
1. Recognize the Sum of Cubes Formula:
The sum of cubes formula states:
[tex]\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \][/tex]
2. Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
Look at the original expression [tex]\(x^3 y^3 + 343\)[/tex]. We need to match it to the sum of cubes formula.
- Notice that [tex]\(x^3 y^3\)[/tex] can be written as [tex]\((xy)^3\)[/tex]. Hence, we can let [tex]\(a = xy\)[/tex].
- Observe that 343 is a perfect cube, specifically [tex]\(7^3\)[/tex]. Therefore, we can let [tex]\(b = 7\)[/tex].
3. Apply the Formula:
- We can now express [tex]\(a\)[/tex] and [tex]\(b\)[/tex] in the sum of cubes formula:
[tex]\[ (xy)^3 + 7^3 = (xy + 7)((xy)^2 - (xy)(7) + 7^2) \][/tex]
4. Simplify the Inner Expression:
- Simplify each component inside the second factor:
- [tex]\((xy)^2 = x^2 y^2\)[/tex]
- [tex]\((xy)(7) = 7xy\)[/tex]
- [tex]\(7^2 = 49\)[/tex]
5. Rewrite the Factors:
- With these simplifications, rewrite the factors as follows:
[tex]\[ (xy + 7)(x^2 y^2 - 7xy + 49) \][/tex]
By identifying the appropriate values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex], and substituting into the sum of cubes formula, we factorized [tex]\(x^3 y^3 + 343\)[/tex].
