Certainly! Let's go through the simplification step by step.
We are given the expression:
[tex]\[ \sqrt{x^{12} y^9 z^5} \][/tex]
First, we rewrite it by breaking down the exponents into perfect squares where possible:
[tex]\[ \sqrt{x^{12} \cdot y^8 \cdot y \cdot z^4 \cdot z} \][/tex]
Now we can separate the square roots of the perfect square terms:
[tex]\[ \sqrt{x^{12}} \cdot \sqrt{y^8} \cdot \sqrt{z^4} \cdot \sqrt{y \cdot z} \][/tex]
Next, we simplify the square roots of each term:
[tex]\[ \sqrt{x^{12}} = x^{12/2} = x^6 \][/tex]
[tex]\[ \sqrt{y^8} = y^{8/2} = y^4 \][/tex]
[tex]\[ \sqrt{z^4} = z^{4/2} = z^2 \][/tex]
Combining these results, we get:
[tex]\[ x^6 \cdot y^4 \cdot z^2 \cdot \sqrt{y \cdot z} \][/tex]
From this, we can identify the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] in the equation [tex]\( x^a y^b z^c \sqrt{y \cdot z} \)[/tex]:
- [tex]\( a = 6 \)[/tex]
- [tex]\( b = 4 \)[/tex]
- [tex]\( c = 2 \)[/tex]
So, the values are:
[tex]\[ a = 6 \][/tex]
[tex]\[ b = 4 \][/tex]
[tex]\[ c = 2 \][/tex]
