Sure, let's solve the problem step-by-step by finding the product in its simplest form:
1. Multiply the Coefficients:
First, consider the coefficients of the terms \(5\) and \(-3\). When multiplying these together:
[tex]\[
5 \cdot (-3) = -15
\][/tex]
2. Add the Exponents:
Next, look at the exponents of \(x\). In the expression \(5x^6 \cdot (-3x^3)\), the exponents are \(6\) and \(3\). When multiplying terms with the same base, you add their exponents:
[tex]\[
x^6 \cdot x^3 = x^{6+3} = x^9
\][/tex]
3. Combine the Results:
Now, combine the result of the multiplied coefficients and the sum of the exponents:
[tex]\[
-15 \cdot x^9 = -15x^9
\][/tex]
Therefore, the product \(5x^6 \cdot (-3x^3)\) in its simplest form is:
[tex]\[
\boxed{-15x^9}
\][/tex]
