Sure, let's tackle the expression step-by-step. We need to simplify the expression [tex]\( 3 y z \left( y^2 z - y z^2 + y z \right) \)[/tex] given [tex]\( y = -2 \)[/tex] and [tex]\( z = 3 \)[/tex].
1. Determine each term inside the parentheses:
- Calculate [tex]\( y^2 z \)[/tex]:
[tex]\[
y^2 z = (-2)^2 \cdot 3 = 4 \cdot 3 = 12
\][/tex]
- Calculate [tex]\( y z^2 \)[/tex]:
[tex]\[
y z^2 = (-2) \cdot 3^2 = (-2) \cdot 9 = -18
\][/tex]
- Calculate [tex]\( y z \)[/tex]:
[tex]\[
y z = (-2) \cdot 3 = -6
\][/tex]
2. Substitute these values back into the inner expression and simplify:
[tex]\[
y^2 z - y z^2 + y z = 12 - (-18) + (-6)
\][/tex]
Simplify the expression inside the parentheses step-by-step:
[tex]\[
12 + 18 - 6 = 24
\][/tex]
So, the inner result is:
[tex]\[
y^2 z - y z^2 + y z = 24
\][/tex]
3. Multiply the simplified inner result by [tex]\(3 y z\)[/tex]:
Calculate [tex]\( 3 y z \)[/tex]:
[tex]\[
3 y z = 3 \cdot (-2) \cdot 3 = 3 \cdot -6 = -18
\][/tex]
Now multiply this by the inner result (24):
[tex]\[
3 y z \left( y^2 z - y z^2 + y z \right) = -18 \cdot 24 = -432
\][/tex]
Hence, the final result is:
[tex]\[
3 y z \left( y^2 z - y z^2 + y z \right) = -432
\][/tex]
So, the detailed, step-by-step solution for the expression [tex]\(3 y z\left(y^2 z-y z^2+y z\right)\)[/tex] given [tex]\(y = -2\)[/tex] and [tex]\(z = 3\)[/tex] reveals that the final result is [tex]\(-432\)[/tex].
