Find the least common multiple of these two expressions. $$ 9 w^{3} y^{5} v^{8} \text { and } 15 w^{2} v^{6} $$

Question

UpStudy AI · Accepted Answer

To find the least common multiple (LCM) of the two expressions $$ 9 w^{3} y^{5} v^{8} $$ and $$ 15 w^{2} v^{6} $$, we will first factor each expression into its prime factors and variables.

1. **Factor the coefficients:**
   - The coefficient of the first expression is $$ 9 $$, which can be factored as:
     $$
     9 = 3^2
     $$
   - The coefficient of the second expression is $$ 15 $$, which can be factored as:
     $$
     15 = 3^1 \times 5^1
     $$

2. **Combine the variable parts:**
   - The first expression has the variables $$ w^3, y^5, v^8 $$.
   - The second expression has the variables $$ w^2, v^6 $$.

3. **Determine the LCM of the coefficients:**
   - For the coefficient $$ 9 = 3^2 $$ and $$ 15 = 3^1 \times 5^1 $$, we take the highest power of each prime factor:
     - For $$ 3 $$: the maximum power is $$ 3^2 $$.
     - For $$ 5 $$: the maximum power is $$ 5^1 $$.
   - Thus, the LCM of the coefficients is:
     $$
     3^2 \times 5^1 = 9 \times 5 = 45
     $$

4. **Determine the LCM of the variable parts:**
   - For $$ w $$: the maximum power is $$ w^3 $$ (from the first expression).
   - For $$ y $$: the maximum power is $$ y^5 $$ (from the first expression).
   - For $$ v $$: the maximum power is $$ v^8 $$ (from the first expression).
   - For $$ w^2 $$: the maximum power is $$ w^2 $$ (from the second expression).
   - Thus, the LCM of the variable parts is:
     $$
     w^3 \cdot y^5 \cdot v^8
     $$

5. **Combine the LCM of the coefficients and the variable parts:**
   - The overall LCM is:
     $$
     45 w^3 y^5 v^8
     $$

Thus, the least common multiple of the two expressions $$ 9 w^{3} y^{5} v^{8} $$ and $$ 15 w^{2} v^{6} $$ is:
$$
\boxed{45 w^3 y^5 v^8}
$$
The least common multiple is $$ 45 w^3 y^5 v^8 $$.

Find the least common multiple of these two expressions. \[ 9 w^{3} y^{5} v^{8} \text { and } 15 w^{2} v^{6} \]

Solución de inteligencia artificial de Upstudy

Responder

Solución

Beyond the Answer

preguntas relacionadas

Latest Algebra Questions