To find the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] in the equation
[tex]\[
\frac{x^5 y z^4 y^3}{x^3 y z} = x^a y^b z^c,
\][/tex]
let's follow these steps:
1. Simplify the expression in the numerator:
[tex]\[
x^5 \cdot y \cdot z^4 \cdot y^3
\][/tex]
We notice that the terms involving [tex]\( y \)[/tex] can be combined since they share the same base:
[tex]\[
y \cdot y^3 = y^{1+3} = y^4
\][/tex]
So, the numerator becomes:
[tex]\[
x^5 \cdot y^4 \cdot z^4
\][/tex]
2. Write the simplified numerator over the denominator:
[tex]\[
\frac{x^5 \cdot y^4 \cdot z^4}{x^3 \cdot y \cdot z}
\][/tex]
3. Simplify each term by dividing the terms in the numerator by their corresponding terms in the denominator:
- For the [tex]\( x \)[/tex] terms:
[tex]\[
\frac{x^5}{x^3} = x^{5-3} = x^2
\][/tex]
- For the [tex]\( y \)[/tex] terms:
[tex]\[
\frac{y^4}{y} = y^{4-1} = y^3
\][/tex]
- For the [tex]\( z \)[/tex] terms:
[tex]\[
\frac{z^4}{z} = z^{4-1} = z^3
\][/tex]
Putting it all together, we get:
[tex]\[
\frac{x^5 \cdot y^4 \cdot z^4}{x^3 \cdot y \cdot z} = x^2 \cdot y^3 \cdot z^3
\][/tex]
Thus, the equation is simplified to:
[tex]\[
x^2 y^3 z^3 = x^a y^b z^c
\][/tex]
From this, we can see that:
[tex]\[
a = 2, \quad b = 3, \quad c = 3
\][/tex]
Therefore, the values are:
[tex]\[
a = 2, \quad b = 3, \quad c = 3
\][/tex]
