https://file.aiquickdraw.com/mathbot/file/1725547842671siyxv80u.jpg

The equation shown in the image is:

\[
y = \sqrt{(2x + 2)^4}
\]

We can simplify this expression step by step.

### Simplification:

1. The square root and the exponent of 4 can be combined:
   \[
   \sqrt{(2x + 2)^4} = (2x + 2)^2
   \]
   This is because \(\sqrt{a^4} = a^2\).

2. Now expand \((2x + 2)^2\):
   \[
   (2x + 2)^2 = (2x + 2)(2x + 2)
   \]

3. Perform the multiplication:
   \[
   (2x + 2)(2x + 2) = 4x^2 + 8x + 4
   \]

Thus, the simplified expression for \(y\) is:
\[
y = 4x^2 + 8x + 4
\]

Let me know if you need further details on this simplification, or if you'd like me to assist you with any other problem!

### Related Questions:
1. How would this expression change if the exponent were different, say \( (2x + 2)^6 \)?
2. What is the derivative of the simplified function \( y = 4x^2 + 8x + 4 \)?
3. How do you solve for \(x\) when \( y = 4x^2 + 8x + 4 \)?
4. Can you factor \( y = 4x^2 + 8x + 4 \)?
5. How would you approach simplifying an expression with a cube root instead?

### Tip:
Whenever you see a square root combined with an even exponent, it's often a hint that the expression can be simplified by reducing the exponent first.

Learn how to simplify the expression y = √((2x + 2)^4) step by step. This problem covers key algebraic concepts, including exponent simplification, square roots, and expansion techniques. Suitable for high school students in Grades 9-12.

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