We are tasked with simplifying the expression:

$\left( \frac{x^6}{x^3} \right)^{-2}$

Step 1: Simplify the expression inside the parentheses using properties of exponents.

Recall the quotient rule for exponents: $\frac{x^a}{x^b} = x^{a - b}$ Applying this rule:

$\frac{x^6}{x^3} = x^{6 - 3} = x^3$

Step 2: Apply the exponent outside the parentheses.

Now, we have: $\left( x^3 \right)^{-2}$ Using the power rule for exponents, $(x^a)^b = x^{a \cdot b}$, we can simplify: $\left( x^3 \right)^{-2} = x^{3 \cdot (-2)} = x^{-6}$

Step 3: Express the answer with positive exponents.

To express $x^{-6}$ with a positive exponent, we use the rule $x^{-a} = \frac{1}{x^a}$: $x^{-6} = \frac{1}{x^6}$

Thus, the simplified expression is:

$\frac{1}{x^6}$

Would you like any further clarification or details on the steps?

Here are 5 related questions for practice:

1. Simplify $\left( \frac{y^8}{y^2} \right)^{-3}$.
2. Simplify $\left( \frac{a^5}{a^2} \right)^{-4}$.
3. Simplify $\left( \frac{b^7}{b^3} \right)^{-1}$.
4. Simplify $\left( \frac{z^{10}}{z^4} \right)^{-2}$.
5. Simplify $\left( \frac{m^9}{m^6} \right)^{-5}$.

Tip: When dealing with negative exponents, always remember to invert the base and change the exponent to positive.