Let's simplify the given expression step by step to write it only using positive exponents.
Given expression:
[tex]\[
\frac{16 x^4 y^{-3} z^4}{36 x^{-2} y z^0}
\][/tex]
### Step 1: Simplify the constants
First, we simplify the numerical coefficients (i.e., the numbers without variables):
[tex]\[
\frac{16}{36} = \frac{4}{9}
\][/tex]
So, our expression becomes:
[tex]\[
\frac{4 x^4 y^{-3} z^4}{9 x^{-2} y z^0}
\][/tex]
### Step 2: Simplify the exponents using properties of exponents
Let's deal with each variable separately.
For variable [tex]\( x \)[/tex]:
We have [tex]\( x^4 \)[/tex] in the numerator and [tex]\( x^{-2} \)[/tex] in the denominator. Recall the property:
[tex]\[
\frac{a^m}{a^n} = a^{m-n}
\][/tex]
Applying this property:
[tex]\[
\frac{x^4}{x^{-2}} = x^{4 - (-2)} = x^{4 + 2} = x^6
\][/tex]
For variable [tex]\( y \)[/tex]:
We have [tex]\( y^{-3} \)[/tex] in the numerator and [tex]\( y \)[/tex] (which is [tex]\( y^1 \)[/tex]) in the denominator:
[tex]\[
\frac{y^{-3}}{y} = y^{-3 - 1} = y^{-3 - 1} = y^{-4}
\][/tex]
For variable [tex]\( z \)[/tex]:
We have [tex]\( z^4 \)[/tex] in the numerator and [tex]\( z^0 \)[/tex] (which is [tex]\( 1 \)[/tex]) in the denominator:
[tex]\[
\frac{z^4}{z^0} = z^4
\][/tex]
### Step 3: Combine all simplified terms
Putting it all together, we get:
[tex]\[
\frac{4 x^6 y^{-4} z^4}{9}
\][/tex]
### Step 4: Write only using positive exponents
To write the expression using only positive exponents, we move [tex]\( y^{-4} \)[/tex] from the numerator to the denominator:
[tex]\[
\frac{4 x^6 z^4}{9 y^4}
\][/tex]
### Final Answer
Thus, the simplified expression using only positive exponents is:
[tex]\[
\frac{4 x^6 z^4}{9 y^4}
\][/tex]
