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**Simplify the following expressions using the law of exponents**

Let's simplify the expression:

\[
2x^3 y^{-3} \cdot 2x^{-1} y^3.
\]

### Step 1: Group similar terms

We can group the coefficients, \(x\)-terms, and \(y\)-terms together:

\[
(2 \cdot 2) \cdot (x^3 \cdot x^{-1}) \cdot (y^{-3} \cdot y^3).
\]

### Step 2: Simplify the coefficients

Multiply the constants:

\[
2 \cdot 2 = 4.
\]

### Step 3: Apply the laws of exponents

For the \(x\)-terms:

\[
x^3 \cdot x^{-1} = x^{3 + (-1)} = x^{2}.
\]

For the \(y\)-terms:

\[
y^{-3} \cdot y^3 = y^{(-3) + 3} = y^0 = 1.
\]

### Step 4: Combine everything

Now, combine all the simplified terms:

\[
4 \cdot x^2 \cdot 1 = 4x^2.
\]

### Final Simplified Expression:
\[
4x^2.
\]

This is the fully simplified expression.

Would you like further clarification on any steps, or need more details?

Here are some related questions to consider:
1. What happens when multiplying terms with the same base but different exponents?
2. How does a term raised to the power of zero simplify in an expression?
3. How do you handle exponents when dividing similar bases?
4. What is the difference between multiplying coefficients and multiplying powers?
5. Can exponent rules be applied to more complex algebraic expressions?

**Tip:** Always combine like terms by adding or subtracting their exponents when they share the same base, and remember that any term raised to the power of zero equals 1.

Learn how to simplify the expression 2x^3 y^-3 * 2x^-1 y^3 using the laws of exponents with a step-by-step solution. This problem involves combining like terms, applying exponent rules, and simplifying algebraic expressions. Suitable for students in Grades 7-9.

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