To simplify the expression [tex]\(\frac{(x y)^3}{9 x}\)[/tex] and ensure it contains only positive exponents, follow these steps:
### Step 1: Expand the Numerator
First, let's look at the numerator [tex]\((xy)^3\)[/tex]. This can be expanded using the property of exponents [tex]\((ab)^n = a^n \cdot b^n\)[/tex]:
[tex]\[
(xy)^3 = x^3 \cdot y^3
\][/tex]
So the numerator becomes [tex]\(x^3 \cdot y^3\)[/tex].
### Step 2: Write the Denominator Clearly
The denominator is already given as:
[tex]\[
9x
\][/tex]
### Step 3: Set Up the Fraction
Now, we can write the fraction in its expanded form:
[tex]\[
\frac{x^3 \cdot y^3}{9x}
\][/tex]
### Step 4: Simplify the Fraction
To simplify, we need to divide the [tex]\(x\)[/tex] terms in the numerator and denominator. Break it down step-by-step:
1. Divide [tex]\(x^3\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{x^3}{x} = x^{3-1} = x^2
\][/tex]
So the simplified fraction now becomes:
[tex]\[
\frac{x^2 \cdot y^3}{9}
\][/tex]
### Step 5: Final Simplified Expression
The expression [tex]\(\frac{x^2 \cdot y^3}{9}\)[/tex] is already in its simplest form with only positive exponents.
So the simplified expression is:
[tex]\[
\boxed{\frac{x^2 \cdot y^3}{9}}
\][/tex]
This is the required simplified form of the given expression with only positive exponents.
