Certainly! Let's dive into the problem step by step.
We need to divide the expression [tex]\( 12 x y^3 z^6 \)[/tex] by [tex]\( 4 x^5 y z^{12} \)[/tex].
### Step 1: Divide the coefficients
First, let's handle the numerical coefficients:
[tex]\[
\frac{12}{4} = 3
\][/tex]
### Step 2: Subtract the exponents for each variable
Next, we subtract the exponents of each corresponding variable in the denominator from those in the numerator.
#### For [tex]\( x \)[/tex]:
The exponent of [tex]\( x \)[/tex] in the numerator is 1, and in the denominator, it is 5. Therefore,
[tex]\[
1 - 5 = -4
\][/tex]
So, the resulting exponent for [tex]\( x \)[/tex] is [tex]\( -4 \)[/tex].
#### For [tex]\( y \)[/tex]:
The exponent of [tex]\( y \)[/tex] in the numerator is 3, and in the denominator, it is 1. Therefore,
[tex]\[
3 - 1 = 2
\][/tex]
So, the resulting exponent for [tex]\( y \)[/tex] is [tex]\( 2 \)[/tex].
#### For [tex]\( z \)[/tex]:
The exponent of [tex]\( z \)[/tex] in the numerator is 6, and in the denominator, it is 12. Therefore,
[tex]\[
6 - 12 = -6
\][/tex]
So, the resulting exponent for [tex]\( z \)[/tex] is [tex]\( -6 \)[/tex].
### Step 3: Write down the result
Combining all these results:
[tex]\[
3 x^{-4} y^2 z^{-6}
\][/tex]
### Step 4: Simplify the expression
Typically, expressions with negative exponents are rewritten in fraction form. We move the terms with negative exponents to the denominator.
So, the final simplified expression is:
[tex]\[
\frac{3 y^2}{x^4 z^6}
\][/tex]
Thus, the result of dividing [tex]\( 12 x y^3 z^6 \)[/tex] by [tex]\( 4 x^5 y z^{12} \)[/tex] is:
[tex]\[
\frac{3 y^2}{x^4 z^6}
\][/tex]
This matches one of the given answer options.
