If `y=3^(x)` and x and y are both integers, which of the following is equivalent to `9^(x)+3^(x+1)`?

Options

A. `y^(3)`
B. `3y+3`
C. `y(y+3)`
D. `y^(2)+3`

Answer

To solve the problem, we need to find an equivalent expression for \( 9^x + 3^{x+1} \) given that \( y = 3^x \).

### Step-by-Step Solution:

1. **Rewrite \( 9^x \)**:
\[ 9^x = (3^2)^x = 3^{2x} \]
This step uses the property of exponents that states \( a^{m^n} = a^{mn} \).

2. **Rewrite \( 3^{x+1} \)**:
\[ 3^{x+1} = 3^x \cdot 3^1 = 3^x \cdot 3 \]
Here, we use the property of exponents that states \( a^{m+n} = a^m \cdot a^n \).

3. **Combine the expressions**:
Now we can combine the two rewritten parts:
\[ 9^x + 3^{x+1} = 3^{2x} + 3^x \cdot 3 \]

4. **Factor out \( 3^x \)**:
\[ 3^{2x} + 3^{x+1} = 3^{2x} + 3^x \cdot 3 = 3^x(3^x + 3) \]
Here, we factor out \( 3^x \) from both terms.

5. **Substitute \( y \)**:
Since \( y = 3^x \), we can substitute \( y \) into the expression:
\[ 3^x(3^x + 3) = y(y + 3) \]

6. **Final Expression**:
Thus, the expression \( 9^x + 3^{x+1} \) can be rewritten as:
\[ y^2 + 3y \]

### Conclusion:
The equivalent expression for \( 9^x + 3^{x+1} \) is \( y^2 + 3y \).