Sure, let's break down the given expression step by step:
Given expression:
[tex]\[ \frac{27 x^5 y^4 z + 21 x^2 y^3 z^2 - 15 x y z}{3 x y z} \][/tex]
Step 1: Identify the terms in the numerator.
The numerator is:
[tex]\[ 27 x^5 y^4 z + 21 x^2 y^3 z^2 - 15 x y z \][/tex]
Step 2: Factor each term in the numerator by their common factors.
Each term in the numerator can be expressed as:
- [tex]\( 27 x^5 y^4 z \)[/tex]
- [tex]\( 21 x^2 y^3 z^2 \)[/tex]
- [tex]\( 15 x y z \)[/tex]
Step 3: Simplify the numerator terms by dividing each by the denominator, [tex]\( 3 x y z \)[/tex].
Let's handle each term separately:
1. [tex]\( \frac{27 x^5 y^4 z}{3 x y z} \)[/tex]:
[tex]\[ \frac{27 x^5 y^4 z}{3 x y z} = \frac{27}{3} \cdot \frac{x^5}{x} \cdot \frac{y^4}{y} \cdot \frac{z}{z} = 9 x^4 y^3 \][/tex]
2. [tex]\( \frac{21 x^2 y^3 z^2}{3 x y z} \)[/tex]:
[tex]\[ \frac{21 x^2 y^3 z^2}{3 x y z} = \frac{21}{3} \cdot \frac{x^2}{x} \cdot \frac{y^3}{y} \cdot \frac{z^2}{z} = 7 x y^2 z \][/tex]
3. [tex]\( \frac{15 x y z}{3 x y z} \)[/tex]:
[tex]\[ \frac{15 x y z}{3 x y z} = \frac{15}{3} \cdot \frac{x}{x} \cdot \frac{y}{y} \cdot \frac{z}{z} = 5 \][/tex]
Step 4: Combine the simplified terms.
After simplifying each term individually, we get:
[tex]\[ 9 x^4 y^3 + 7 x y^2 z - 5 \][/tex]
Thus, the simplified expression is:
[tex]\[ 9 x^4 y^3 + 7 x y^2 z - 5 \][/tex]
