Let's solve the given multiplication and simplify the expression step-by-step:
Given expression:
[tex]\[ 2u^6x^5 \cdot 4x^9 \cdot 6u \][/tex]
Step 1: Multiply the coefficients
First, multiply the numerical coefficients:
[tex]\[ 2 \cdot 4 \cdot 6 = 48 \][/tex]
Step 2: Combine the variables with their respective exponents
Next, multiply the variables with like terms.
For \( u \)-terms:
[tex]\[ u^6 \cdot u = u^6 \cdot u^1 \][/tex]
When multiplying terms with the same base, add their exponents:
[tex]\[ 6 + 1 = 7 \][/tex]
So, \( u^6 \cdot u = u^7 \)
For \( x \)-terms:
[tex]\[ x^5 \cdot x^9 = x^{5+9} \][/tex]
Similarly, add the exponents:
[tex]\[ 5 + 9 = 14 \][/tex]
So, \( x^5 \cdot x^9 = x^{14} \)
Step 3: Combine all the results
Now, combine the results from steps 1 and 2:
[tex]\[ 48 u^7 x^{14} \][/tex]
Final Simplified Expression:
[tex]\[
48u^7x^{14}
\][/tex]
So, the simplified form of the expression \( 2u^6x^5 \cdot 4x^9 \cdot 6u \) is:
[tex]\[
\boxed{48u^7x^{14}}
\][/tex]
