1. **State the problem:** Find the least common multiple (LCM) of the expressions $12w^4 y^7$ and $10w^6 y^2 x^8$. 2. **Recall the formula and rules:** The LCM of algebraic expressions involves taking the highest powers of each variable and the least common multiple of the coefficients. 3. **Find the LCM of the coefficients:** The coefficients are 12 and 10. Prime factorization: $$12 = 2^2 \times 3$$ $$10 = 2 \times 5$$ LCM of coefficients is the product of the highest powers of all prime factors: $$LCM(12,10) = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60$$ 4. **Find the LCM of the variables:** - For $w$, powers are $w^4$ and $w^6$. Take the highest power: $w^6$. - For $y$, powers are $y^7$ and $y^2$. Take the highest power: $y^7$. - For $x$, it appears only in the second expression as $x^8$. Include it as is: $x^8$. 5. **Combine the LCM of coefficients and variables:** $$LCM = 60 w^6 y^7 x^8$$ 6. **Final answer:** The least common multiple of $12w^4 y^7$ and $10w^6 y^2 x^8$ is $$\boxed{60 w^6 y^7 x^8}$$