\left(\left(3\sqrt{2}\right)\left(5\sqrt{3}\right)\right)^2+\left(\frac{1}{2}\right)^{-2}

Let's break down the expression step by step:

The expression is:

\[
\left(\left(3\sqrt{2}\right)\left(5\sqrt{3}\right)\right)^2 + \left(\frac{1}{2}\right)^{-2}
\]

### Step 1: Simplify the product inside the first parenthesis
Multiply \(3\sqrt{2}\) and \(5\sqrt{3}\):

\[
3\sqrt{2} \times 5\sqrt{3} = (3 \times 5)(\sqrt{2} \times \sqrt{3}) = 15\sqrt{6}
\]

### Step 2: Square the result
Square the expression \(15\sqrt{6}\):

\[
(15\sqrt{6})^2 = 15^2 \times (\sqrt{6})^2 = 225 \times 6 = 1350
\]

### Step 3: Simplify the second term
Simplify \(\left(\frac{1}{2}\right)^{-2}\):

\[
\left(\frac{1}{2}\right)^{-2} = \left(\frac{2}{1}\right)^2 = 2^2 = 4
\]

### Step 4: Add the results
Now, add the two results together:

\[
1350 + 4 = 1354
\]

So, the final answer is:

\[
\boxed{1354}
\]

Would you like further details or have any questions? Here are 5 questions that could extend this problem:

1. What would happen if the exponents in the problem were different?
2. How would the problem change if the constants inside the square roots were different?
3. Can you factorize the final result \(1354\) into prime factors?
4. How would the expression change if \(\frac{1}{2}\) was raised to a different negative power?
5. What if instead of squaring, we had to take the square root of \((15\sqrt{6})\)?

**Tip:** Always simplify expressions inside parentheses first to make complex problems more manageable.

Learn how to solve the mathematical expression (3√2 * 5√3)^2 + (1/2)^(-2) step-by-step with detailed explanations. This problem involves exponentiation and operations with square roots, suitable for advanced high school students.

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