To simplify the expression [tex]\( 8v \cdot 3u^8 \cdot 2v^2u^4 \)[/tex], follow these steps:
1. Multiply the numerical coefficients:
- The numerical coefficients in the expression are [tex]\( 8 \)[/tex], [tex]\( 3 \)[/tex], and [tex]\( 2 \)[/tex].
- To find the product of these coefficients, multiply them together:
[tex]\[
8 \times 3 \times 2 = 48
\][/tex]
2. Combine the exponents of the variable [tex]\( u \)[/tex]:
- The expression has [tex]\( u \)[/tex] raised to different powers: [tex]\( u^8 \)[/tex] and [tex]\( u^4 \)[/tex].
- Use the laws of exponents, which state that when you multiply powers of the same base, you add the exponents:
[tex]\[
u^8 \cdot u^4 = u^{8 + 4} = u^{12}
\][/tex]
3. Combine the exponents of the variable [tex]\( v \)[/tex]:
- The expression has [tex]\( v \)[/tex] raised to different powers: [tex]\( v \)[/tex] and [tex]\( v^2 \)[/tex].
- Recall that [tex]\( v \)[/tex] is equivalent to [tex]\( v^1 \)[/tex].
- Similar to the previous step, add the exponents:
[tex]\[
v^1 \cdot v^2 = v^{1 + 2} = v^3
\][/tex]
4. Combine all the parts together:
- Bring together the coefficient with the simplified powers of [tex]\( u \)[/tex] and [tex]\( v \)[/tex]:
[tex]\[
48 \cdot u^{12} \cdot v^3
\][/tex]
Therefore, the simplified expression is:
[tex]\[
48u^{12}v^3
\][/tex]
