https://file.aiquickdraw.com/mathbot/file/1725620844903yp7ktgfh.jpg

The problem in the image appears to be about simplifying an expression. The expression is:

\[
\frac{(x^3y^{-2})^3}{6x^{-9}}
\]

You are required to simplify this expression without using negative exponents. Let's break this down step by step:

### Step 1: Apply the exponent rule \((a^m)^n = a^{mn}\)
For the numerator \((x^3y^{-2})^3\), distribute the exponent \(3\) to both \(x^3\) and \(y^{-2}\):

\[
(x^3y^{-2})^3 = x^{3 \times 3}y^{-2 \times 3} = x^9y^{-6}
\]

So, the expression becomes:

\[
\frac{x^9y^{-6}}{6x^{-9}}
\]

### Step 2: Simplify the exponents of \(x\)
Now, apply the quotient rule \(a^m / a^n = a^{m-n}\) to simplify \(x^9 / x^{-9}\):

\[
\frac{x^9}{x^{-9}} = x^{9 - (-9)} = x^{9 + 9} = x^{18}
\]

So the expression is now:

\[
\frac{x^{18}y^{-6}}{6}
\]

### Step 3: Convert negative exponent to positive
Finally, to remove the negative exponent on \(y^{-6}\), write it as:

\[
\frac{x^{18}}{6y^6}
\]

This is the simplified expression without negative exponents.

The final answer is:

\[
\boxed{\frac{x^{18}}{6y^6}}
\]

---

Would you like further details or have any questions about this?

Here are 5 related questions:
1. How do you simplify expressions with fractional exponents?
2. What is the general rule for dividing terms with exponents?
3. How do you simplify expressions with multiple variables?
4. What happens if the base of an exponent is negative?
5. How do exponent laws apply to roots and radicals?

**Tip:** When simplifying expressions with exponents, always apply the exponent rules carefully, step by step, and watch out for negative exponents!

Simplify the expression: \(\frac{(x^3y^{-2})^3}{6x^{-9}}\). Write your answer without negative exponents.

Learn how to simplify the expression \(\frac{(x^3y^{-2})^3}{6x^{-9}}\) without negative exponents. This step-by-step guide covers exponent rules such as the power rule and quotient rule. Suitable for students in Grades 9-11.

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