To solve the expression [tex]\( 3x \cdot 2x^3 u^9 \cdot 3u^8 \)[/tex], we need to follow a systematic approach by grouping and simplifying the terms step-by-step. Here's the detailed solution:
### Step 1: Group Similar Terms
First, we'll identify and group the similar terms, which are those containing [tex]\(x\)[/tex] and those containing [tex]\(u\)[/tex].
#### Terms involving [tex]\( x \)[/tex]:
[tex]\[ 3x \cdot 2x^3 \][/tex]
#### Terms involving [tex]\( u \)[/tex]:
[tex]\[ u^9 \cdot 3u^8 \][/tex]
### Step 2: Simplify the Terms Involving [tex]\( x \)[/tex]
Multiply the coefficients and the variables separately.
- Coefficients: [tex]\( 3 \times 2 = 6 \)[/tex]
- Variables: [tex]\( x \times x^3 = x^{1+3} = x^4 \)[/tex]
So, the simplified form for the terms involving [tex]\( x \)[/tex] is:
[tex]\[ (3x) \cdot (2x^3) = 6x^4 \][/tex]
### Step 3: Simplify the Terms Involving [tex]\( u \)[/tex]
Multiply the coefficients and the variables separately.
- Coefficient: The only coefficient here is [tex]\(3\)[/tex] since [tex]\(u^9\)[/tex] and [tex]\(u^8\)[/tex] have implicit coefficients of 1.
- Variables: [tex]\( u^9 \cdot u^8 = u^{9+8} = u^{17} \)[/tex]
So, the simplified form for the terms involving [tex]\( u \)[/tex] is:
[tex]\[ u^9 \cdot 3u^8 = 3u^{17} \][/tex]
### Step 4: Combine the Simplified Terms
Now, multiply the results of the simplified terms involving [tex]\( x \)[/tex] and [tex]\( u \)[/tex]:
[tex]\[ 6x^4 \cdot 3u^{17} \][/tex]
Multiply the coefficients and then write the variables together:
- Coefficients: [tex]\( 6 \times 3 = 18 \)[/tex]
So the combined simplified expression is:
[tex]\[ 6x^4 \cdot 3u^{17} = 18x^4 u^{17} \][/tex]
### Final Answer
[tex]\[ 3x \cdot 2x^3 u^9 \cdot 3u^8 = 18x^4 u^{17} \][/tex]
Thus, the simplified form of the given expression is [tex]\( 18x^4 u^{17} \)[/tex].
