Let's simplify each expression step-by-step.
### Expression 1: [tex]\(3.14 a b^3 \times (-3 a^3 b)\)[/tex]
1. Combine the coefficients:
[tex]\[
3.14 \times (-3) = -9.42
\][/tex]
2. Combine the [tex]\(a\)[/tex] terms:
[tex]\[
a \times a^3 = a^{1+3} = a^4
\][/tex]
3. Combine the [tex]\(b\)[/tex] terms:
[tex]\[
b^3 \times b = b^{3+1} = b^4
\][/tex]
4. Put it all together:
[tex]\[
-9.42 a^4 b^4
\][/tex]
The simplified form of Expression 1 is:
[tex]\[
-9.42 a^4 b^4
\][/tex]
### Expression 2: [tex]\(3.2 \frac{2 x^3 + 8 x^3}{(5 x)(2 x)^2}\)[/tex]
1. Simplify the numerator:
[tex]\[
2 x^3 + 8 x^3 = 10 x^3
\][/tex]
2. Simplify the denominator:
First, simplify [tex]\((2 x)^2\)[/tex]:
[tex]\[
(2 x)^2 = 4 x^2
\][/tex]
Then, multiply it by [tex]\(5 x\)[/tex]:
[tex]\[
(5 x)(4 x^2) = 5 x \times 4 x^2 = 20 x^3
\][/tex]
3. Form the fraction and simplify:
[tex]\[
\frac{10 x^3}{20 x^3} = \frac{10}{20} = \frac{1}{2}
\][/tex]
4. Multiply by 3.2:
[tex]\[
3.2 \times \frac{1}{2} = 1.6
\][/tex]
The simplified form of Expression 2 is:
[tex]\[
1.6
\][/tex]
### Final Simplified Results
- Expression 1: [tex]\(-9.42 a^4 b^4\)[/tex]
- Expression 2: [tex]\(1.6\)[/tex]
These are the simplified forms for the given mathematical expressions.
