To simplify $7\sqrt{600}$:

Step 1: Factor $600$ into its prime factors.

$600 = 2^3 \times 3 \times 5^2$

Step 2: Apply the square root rule:

$\sqrt{600} = \sqrt{2^3 \times 3 \times 5^2}$

Step 3: Simplify by taking square factors out of the square root:

$\sqrt{600} = \sqrt{(2^2 \times 5^2) \times (2 \times 3)} = 2 \times 5 \times \sqrt{6} = 10\sqrt{6}$

Step 4: Multiply by 7:

$7\sqrt{600} = 7 \times 10\sqrt{6} = 70\sqrt{6}$

Final Answer:

$\boxed{70\sqrt{6}}$

Would you like a breakdown of why certain steps work? Here are related questions for practice:

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1. Simplify $9\sqrt{150}$.
2. Simplify $5\sqrt{72}$.
3. What is the prime factorization of $600$?
4. How do you know when to pull factors outside of a square root?
5. Compute $3\sqrt{24}$.

Tip: Remember, pulling perfect squares out of a radical is key to simplifying expressions efficiently.