Which statements is true for all real values of x and y?

Options

A. `(x+y)^(2)=x^(2)+y^(2)`
B. `x^(2)+x^(2)=x^(4)`
C. `(2^(x+2))/(2^(x))=4`
D. `(2^(x+2))/(2^(x))=-4`

Answer

To determine which statement is true for all real values of \( x \) and \( y \), we need to analyze each option provided.

### Step-by-Step Solution:

1. **Evaluate Option A**:
- The statement is \( (x + y)^2 = x^2 + y^2 \).
- Expanding the left side using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \):
\[ (x + y)^2 = x^2 + 2xy + y^2 \]
- This shows that \( (x + y)^2 \) is not equal to \( x^2 + y^2 \) unless \( xy = 0 \), which is not true for all real \( x \) and \( y \).
- **Conclusion**: Option A is incorrect.

2. **Evaluate Option B**:
- The statement is \( x^2 + x^2 = x^4 \).
- Simplifying the left side:
\[ 2x^2 \neq x^4 \text{ for all } x \text{ (e.g., for } x = 1, 2, etc.) \]
- **Conclusion**: Option B is incorrect.

3. **Evaluate Option C**:
- The statement is \( 2^{x+2} / 2^x = 4 \).
- Using the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \):
\[ \frac{2^{x+2}}{2^x} = 2^{(x+2) - x} = 2^2 = 4 \]
- This statement holds true for all real values of \( x \).
- **Conclusion**: Option C is correct.

4. **Evaluate Option D**:
- The statement is \( 2^{x+2} / 2^x = -4 \).
- From our evaluation of Option C, we know \( 2^{x+2} / 2^x = 4 \), which is not equal to \(-4\).
- **Conclusion**: Option D is incorrect.

### Final Answer:
The true statement for all real values of \( x \) and \( y \) is **Option C**: \( \frac{2^{x+2}}{2^x} = 4 \).