Yes, the equality \(|(a + b)^2| = |(a + b)|^2\) is true. Here's why:

### Explanation:
1. \((a + b)^2\) refers to squaring the sum of \(a\) and \(b\), which is \((a + b)(a + b)\). The square of any real number (or expression) is non-negative.
   
2. The absolute value of \((a + b)^2\), i.e., \(|(a + b)^2|\), is the distance of \((a + b)^2\) from zero. Since \((a + b)^2\) is always non-negative, \(|(a + b)^2| = (a + b)^2\).

3. On the right side, \(|(a + b)|^2\) refers to squaring the absolute value of \(a + b\). The absolute value of a number is always non-negative, and squaring a non-negative number also gives a non-negative result. So, \(|(a + b)|^2 = (|a + b|)^2\).

Since both sides are non-negative and represent the square of the same expression, \((a + b)^2\), the equality holds true.

Thus:
\[
|(a + b)^2| = |(a + b)|^2
\]

Would you like further details or have any other questions?

### Related Questions:
1. What is the geometric interpretation of absolute value?
2. How does squaring affect negative numbers in absolute value terms?
3. Can \((a + b)^2\) ever be negative? Why not?
4. How does this equality change for complex numbers \(a\) and \(b\)?
5. How does the distributive property work when expanding \((a + b)^2\)?

### Tip:
Remember, squaring a number always results in a non-negative value, even if the original number is negative.

Explore why the equality |(a + b)^2| = |(a + b)|^2 holds true with a detailed explanation. This problem covers key algebraic concepts, including the properties of absolute values and squaring. Suitable for high school students in Grades 9-12.

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