To simplify the given expression $\sqrt{7} (\sqrt{3} + 7)$, let's expand it step by step:

Step 1: Distribute $\sqrt{7}$ to each term inside the parentheses

$\sqrt{7} (\sqrt{3} + 7) = \sqrt{7} \cdot \sqrt{3} + \sqrt{7} \cdot 7.$

Step 2: Simplify each term

1. $\sqrt{7} \cdot \sqrt{3} = \sqrt{21}$ (since $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$).
2. $\sqrt{7} \cdot 7 = 7\sqrt{7}$.

Thus, the expression becomes: $\sqrt{21} + 7\sqrt{7}.$

Step 3: Match the answer with the choices

From the options:

• Option D matches our result: $\sqrt{21} + 7\sqrt{7}$.

Final Answer:

D. $\sqrt{21} + 7\sqrt{7}$.

Do you need a detailed explanation for any step? Let me know!

Related Questions:

1. How do you simplify expressions involving square roots and constants?
2. What is the rule for multiplying square roots $\sqrt{a} \cdot \sqrt{b}$?
3. How do you distribute a square root term across parentheses?
4. How would the expression change if $\sqrt{7}$ were squared instead?
5. How can you combine like terms when square roots are involved?

Tip: Always simplify radicals (square roots) first when possible, and look for patterns to combine terms efficiently.