Use properties of exponents to simplify the expression. Express the answer in exponential form with positive exponents only. Assume that any variable in the denominator is not equal to zero.
left parenthesis StartFraction x Superscript 6 Over x cubed EndFraction right parenthesis Superscript negative 2
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Part 1
left parenthesis StartFraction x Superscript 6 Over x cubed EndFraction right parenthesis Superscript negative 2
equals
  
enter your response here 
​(Simplify your answer. Type exponential notation with positive​ exponents.)

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.

Use properties of exponents to simplify the expression. Express the answer in exponential form with positive exponents only. Assume that any variable in the denominator is not equal to zero. (x^6 / x^3)^-2

Learn how to simplify the exponential expression (x^6 / x^3)^-2 using the properties of exponents, including the quotient and power rules. This step-by-step solution expresses the answer with positive exponents and is suitable for students in Grades 8-10.

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