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.