### Problem 3.1: Simplify [tex]\(4ab^3 \times -3a^3b\)[/tex]
To simplify the expression, follow these steps:
1. Multiply the coefficients: Multiply the numerical coefficients together.
[tex]\[
4 \times -3 = -12
\][/tex]
2. Multiply the 'a' terms: Use the exponent addition rule [tex]\(a^m \times a^n = a^{m+n}\)[/tex].
[tex]\[
a \times a^3 = a^{1+3} = a^4
\][/tex]
3. Multiply the 'b' terms: Again, use the exponent addition rule [tex]\(b^m \times b^n = b^{m+n}\)[/tex].
[tex]\[
b^3 \times b = b^{3+1} = b^4
\][/tex]
4. Combine all results:
[tex]\[
4ab^3 \times -3a^3b = -12a^4b^4
\][/tex]
So, the simplified form of [tex]\(4ab^3 \times -3a^3b\)[/tex] is:
[tex]\[
-12a^4b^4
\][/tex]
### Problem 3.2: Simplify [tex]\(\frac{6x^3 + 12x^3}{(3x)(2x^2)}\)[/tex]
To simplify the given rational expression, follow these steps:
1. Combine like terms in the numerator:
[tex]\[
6x^3 + 12x^3 = (6 + 12)x^3 = 18x^3
\][/tex]
2. Simplify the denominator:
[tex]\[
(3x)(2x^2) = 3 \times 2 \times x \times x^2 = 6x^3
\][/tex]
3. Divide the simplified numerator by the simplified denominator:
[tex]\[
\frac{18x^3}{6x^3}
\][/tex]
4. Cancel out the common factors in the numerator and denominator:
[tex]\[
\frac{18}{6} \cdot \frac{x^3}{x^3} = 3 \cdot 1 = 3
\][/tex]
So, the simplified form of [tex]\(\frac{6x^3 + 12x^3}{(3x)(2x^2)}\)[/tex] is:
[tex]\[
3
\][/tex]
### Final Answers:
1. [tex]\(4ab^3 \times -3a^3b = -12a^4b^4\)[/tex]
2. [tex]\(\frac{6x^3 + 12x^3}{(3x)(2x^2)} = 3\)[/tex]
