Let's solve the given problem:
The expression is:

$4\sqrt{28z} + \sqrt{63z}$

Step-by-step Simplification:

1. Simplify the square roots:

• Break down the radicands ($28z$ and $63z$) into their prime factors and perfect squares:

  $\sqrt{28z} = \sqrt{4 \cdot 7z} = \sqrt{4} \cdot \sqrt{7z} = 2\sqrt{7z}.$

  $\sqrt{63z} = \sqrt{9 \cdot 7z} = \sqrt{9} \cdot \sqrt{7z} = 3\sqrt{7z}.$

2. Substitute these simplified forms back into the expression:

$4\sqrt{28z} + \sqrt{63z} = 4(2\sqrt{7z}) + 3\sqrt{7z}.$

3. Multiply and combine like terms:

• Multiply $4$ by $2\sqrt{7z}$: $4(2\sqrt{7z}) = 8\sqrt{7z}.$

• Add $8\sqrt{7z} + 3\sqrt{7z}$: $8\sqrt{7z} + 3\sqrt{7z} = 11\sqrt{7z}.$

Final Answer:

$\boxed{11\sqrt{7z}}$

Let me know if you'd like a further breakdown or clarification! 😊

Related Questions:

1. What is the general rule for simplifying square roots with variables?
2. How do you handle combining like terms involving square roots?
3. How can you identify perfect square factors of a number?
4. What happens if the terms under the square root cannot be simplified?
5. Can square roots be simplified when the variable has an exponent greater than 1?

Tip:

Always factor the radicand completely to look for perfect squares when simplifying square roots!