To simplify this expression involving radicals, we proceed by simplifying each square root term:

The expression given is: $4\sqrt{7} + \sqrt{28} - \sqrt{63}$

Step 1: Simplify each square root term

1. $\sqrt{28}$: $\sqrt{28} = \sqrt{4 \times 7} = \sqrt{4} \times \sqrt{7} = 2\sqrt{7}$

2. $\sqrt{63}$: $\sqrt{63} = \sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7} = 3\sqrt{7}$

Step 2: Substitute and combine terms

Now we substitute back into the expression: $4\sqrt{7} + 2\sqrt{7} - 3\sqrt{7}$

Combine the terms by adding and subtracting the coefficients of $\sqrt{7}$: $(4 + 2 - 3)\sqrt{7} = 3\sqrt{7}$

Final Answer

$4\sqrt{7} + \sqrt{28} - \sqrt{63} = 3\sqrt{7}$

Would you like more details or have any questions?

Here are five questions that might extend your understanding:

1. How do you simplify other square roots in similar expressions?
2. Can you combine terms with different radicands (like $\sqrt{7}$ and $\sqrt{5}$)? Why or why not?
3. What is the general approach for simplifying square roots with larger numbers?
4. How would the result change if the coefficients in the expression were negative?
5. What steps would you take to expand expressions with radicals in a product format?

Tip: When simplifying square roots, always look for factors that are perfect squares to reduce the expression efficiently.