Sure! Let's go through the steps to simplify and evaluate the given expression:
### Step 1: Simplifying the expression
We start with the expression:
[tex]\[
(16 a^3 b^2 c) \times \left(\frac{1}{8} a b\right) \times \left(\frac{1}{4} a^2 b c\right)
\][/tex]
First, we will combine the constants and the variables separately.
#### Combining Constants:
[tex]\[
16 \times \frac{1}{8} \times \frac{1}{4} = 16 \times \frac{1}{32} = \frac{16}{32} = \frac{1}{2}
\][/tex]
#### Combining Variables:
Next, we combine the variable parts.
[tex]\[
a^3 \times a \times a^2 = a^{3+1+2} = a^6
\][/tex]
[tex]\[
b^2 \times b \times b = b^{2+1+1} = b^4
\][/tex]
[tex]\[
c \times c = c^2
\][/tex]
Now, we can put everything together:
[tex]\[
\left(16 a^3 b^2 c\right) \times \left(\frac{1}{8} a b\right) \times \left(\frac{1}{4} a^2 b c\right) = \frac{1}{2} \times a^6 \times b^4 \times c^2
\][/tex]
So the simplified expression is:
[tex]\[
\frac{1}{2} a^6 b^4 c^2
\][/tex]
### Step 2: Evaluating the simplified expression
Now we need to evaluate [tex]\(\frac{1}{2} a^6 b^4 c^2\)[/tex] for [tex]\(a = -1\)[/tex], [tex]\(b = 1\)[/tex], and [tex]\(c = -2\)[/tex].
Substitute these values into the simplified expression:
[tex]\[
\frac{1}{2} (-1)^6 (1)^4 (-2)^2
\][/tex]
Calculate the exponents:
[tex]\[
(-1)^6 = 1 \quad (\text{since } (-1) \text{ raised to an even power is 1})
\][/tex]
[tex]\[
(1)^4 = 1
\][/tex]
[tex]\[
(-2)^2 = 4
\][/tex]
Now, substitute these results back into the expression:
[tex]\[
\frac{1}{2} \times 1 \times 1 \times 4 = \frac{1}{2} \times 4 = \frac{4}{2} = 2
\][/tex]
Therefore, the value of the expression when [tex]\(a = -1\)[/tex], [tex]\(b = 1\)[/tex], and [tex]\(c = -2\)[/tex] is:
[tex]\[
2
\][/tex]
### Final Result
The simplified expression is:
[tex]\[
\frac{1}{2} a^6 b^4 c^2
\][/tex]
And its value for [tex]\(a = -1\)[/tex], [tex]\(b = 1\)[/tex], and [tex]\(c = -2\)[/tex] is:
[tex]\[
2
\][/tex]
