https://file.aiquickdraw.com/mathbot/file/1736475918152x7h5gs5m.jpg

I see the image includes two mathematical expressions:

5. \( \left( 4 \sqrt{a}(x-1) \right)^2 \)  
6. \( \left( a^3 - 6^3 \right)^2 \)

Let me solve them step by step:

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### Problem 5:  
Simplify \( \left( 4 \sqrt{a}(x-1) \right)^2 \).

**Solution**:
1. Expand the square:
   \[
   \left( 4 \sqrt{a}(x-1) \right)^2 = \left( 4 \sqrt{a} \right)^2 \cdot (x-1)^2
   \]

2. Calculate \( \left( 4 \sqrt{a} \right)^2 \):
   \[
   \left( 4 \sqrt{a} \right)^2 = 16a
   \]

3. Combine results:
   \[
   \left( 4 \sqrt{a}(x-1) \right)^2 = 16a (x-1)^2
   \]

**Final Answer**:
\[
16a (x-1)^2
\]

---

### Problem 6:  
Simplify \( \left( a^3 - 6^3 \right)^2 \).

**Solution**:
1. Recall the formula for the cube of a number: \( 6^3 = 216 \).  
   Substituting this:
   \[
   \left( a^3 - 6^3 \right)^2 = \left( a^3 - 216 \right)^2
   \]

2. This is already simplified unless more expansion is required.

**Final Answer**:
\[
\left( a^3 - 216 \right)^2
\]

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Let me know if you want further steps, detailed explanations, or additional examples.

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### Relative Questions:
1. How do you expand expressions like \( (x+y)^2 \) or \( (x-y)^2 \)?
2. Can you explain the difference between squaring and taking the square root?
3. How can \( a^3 - b^3 \) be factored further using the difference of cubes formula?
4. What happens if \( a \) is negative or \( a = 0 \) in these equations?
5. How can this knowledge of expansion apply to physics or engineering problems?

### Tip:
When squaring a term with multiple factors, square each factor individually, then multiply them.

Simplify the expressions (4√a(x-1))^2 and (a^3 - 6^3)^2.

Learn how to simplify algebraic expressions, including (4√a(x-1))^2 and (a^3 - 6^3)^2. Explore step-by-step solutions using exponents, square roots, and polynomial simplification. Suitable for Grades 9-11.

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