Divide [tex]$12 x y^3 z^6$[/tex] by [tex]$4 x^5 y z^{12}$[/tex].

A. [tex]\frac{3 x^5 y^3}{x^5}[/tex]
B. [tex]3 x^{-4} y^2 z^6[/tex]
C. [tex]3 x^{-5} y^3 z^6[/tex]
D. [tex]\frac{3 y^2}{x^4 z^6}[/tex]



Answer :

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.