	Simplify: (〖8x〗^1/(2^2 x^2 y^0 ))^2 (x^5/x^3 )^2	

To simplify the given expression:

\[
\left(\frac{8x^{-1}}{2^2 x^2 y^0}\right)^{-2} \left(\frac{x^5}{x^3}\right)^{-2}
\]

let's simplify each part step by step.

### Step 1: Simplify each component

1. **Simplify \(\left(\frac{8x^{-1}}{2^2 x^2 y^0}\right)^{-2}\):**

   \[
   \frac{8x^{-1}}{2^2 x^2 y^0} = \frac{8x^{-1}}{4x^2}
   \]

   Since \(y^0 = 1\), we can ignore it.

   Now, simplify the fraction:

   \[
   \frac{8x^{-1}}{4x^2} = \frac{8}{4} \cdot \frac{x^{-1}}{x^2} = 2 \cdot x^{-3}
   \]

   Applying the negative exponent rule \((a^m)^n = a^{mn}\):

   \[
   (2x^{-3})^{-2} = 2^{-2} \cdot (x^{-3})^{-2}
   \]

   \[
   = \frac{1}{4} \cdot x^6 = \frac{x^6}{4}
   \]

2. **Simplify \(\left(\frac{x^5}{x^3}\right)^{-2}\):**

   Using the quotient rule \(a^m / a^n = a^{m-n}\):

   \[
   \frac{x^5}{x^3} = x^{5-3} = x^2
   \]

   Now, apply the negative exponent:

   \[
   (x^2)^{-2} = x^{-4}
   \]

### Step 2: Combine the simplified components

Now, multiply the two results:

\[
\frac{x^6}{4} \cdot x^{-4}
\]

Combine the exponents using the rule \(a^m \cdot a^n = a^{m+n}\):

\[
\frac{x^{6-4}}{4} = \frac{x^2}{4}
\]

### Final Simplified Expression

The simplified form of the original expression is:

\[
\frac{x^2}{4}
\]

Would you like more details, or do you have any questions?

Here are some related questions to explore:

1. How do you simplify an expression with multiple exponents?
2. What are the rules for multiplying and dividing powers with the same base?
3. How do negative exponents affect the simplification of fractions?
4. What happens when a variable is raised to the power of zero?
5. How do you handle expressions with both coefficients and variables when simplifying?

**Tip:** When simplifying expressions with exponents, always start by applying the exponent rules such as product of powers, quotient of powers, and power of a power, while keeping an eye on negative exponents and zero exponents.

Simplify: (〖8x〗^1/(2^2 x^2 y^0 ))^2 (x^5/x^3 )^2

Simplify the expression (〖8x〗^1/(2^2 x^2 y^0 ))^2 (x^5/x^3 )^2 using exponent rules. This problem covers key algebraic concepts, including negative exponents, the quotient rule, and the product of powers. Suitable for students in Grades 9-11.

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