To simplify and evaluate the given expression, we follow these steps:
### Step-by-Step Solution:
1. Identify the variables and constants:
We have three variables: [tex]\( p \)[/tex], [tex]\( q \)[/tex], and [tex]\( r \)[/tex]. The expression is:
[tex]\[
pqr(2p^2 - 3pr^2 + q^2r^2)
\][/tex]
2. Understand the structure of the expression:
The given expression consists of a product of [tex]\( pqr \)[/tex] and a polynomial in [tex]\( p \)[/tex], [tex]\( q \)[/tex], and [tex]\( r \)[/tex].
3. Break down the polynomial within parentheses:
The polynomial inside the parentheses is:
[tex]\[
2p^2 - 3pr^2 + q^2r^2
\][/tex]
4. Distribute [tex]\( pqr \)[/tex] through the polynomial:
Multiplying [tex]\( pqr \)[/tex] with each term inside the parentheses gives:
[tex]\[
pqr \cdot 2p^2 - pqr \cdot 3p r^2 + pqr \cdot q^2 r^2
\][/tex]
5. Simplify each term individually:
- The first term:
[tex]\[
pqr \cdot 2p^2 = 2p^3qr
\][/tex]
- The second term:
[tex]\[
pqr \cdot 3pr^2 = 3p^2qr^3
\][/tex]
- The third term:
[tex]\[
pqr \cdot q^2r^2 = pqr^3q^2 = p^1 \cdot q^3 \cdot r^3 = pq^3r^3
\][/tex]
6. Combine all the terms:
Now, put together the terms we obtained:
[tex]\[
2p^3qr - 3p^2qr^3 + pq^3r^3
\][/tex]
### Final Simplified Expression:
The simplified form of the given expression is:
[tex]\[
pqr(2p^2 - 3pr^2 + q^2r^2) = 2p^3qr - 3p^2qr^3 + pq^3r^3
\][/tex]
This is the detailed step-by-step solution for simplifying the mathematical expression provided.
