IF `y=3^(x)`, which of the following expressions is equivalent to `9^(x)-3^(x+2` for all positive integer value of x?

Options

A. `3y-3`
B. `y^(2)-y`
C. `y^(2)-3y`
D. `y^(2)-9y`

Answer

To solve the expression \( 9^x - 3^{x+2} \) given that \( y = 3^x \), we can follow these steps:

### Step 1: Rewrite \( 9^x \)
We know that \( 9 \) can be expressed as \( 3^2 \). Therefore, we can rewrite \( 9^x \) as:
\[ 9^x = (3^2)^x = 3^{2x} \]

### Step 2: Rewrite \( 3^{x+2} \)
Next, we can rewrite \( 3^{x+2} \) as:
\[ 3^{x+2} = 3^x \cdot 3^2 = 3^x \cdot 9 \]

### Step 3: Substitute the expressions into the original equation
Now, we can substitute these expressions back into the original equation:
\[ 9^x - 3^{x+2} = 3^{2x} - 3^x \cdot 9 \]

### Step 4: Factor out \( 3^x \)
We can factor \( 3^x \) out of the expression:
\[ 3^{2x} - 3^x \cdot 9 = 3^x(3^x - 9) \]

### Step 5: Substitute \( y \) for \( 3^x \)
Since we have \( y = 3^x \), we can substitute \( y \) into the equation:
\[ 3^x(3^x - 9) = y(y - 9) \]

### Step 6: Final expression
Thus, the expression \( 9^x - 3^{x+2} \) simplifies to:
\[ y^2 - 9y \]

### Conclusion
The equivalent expression for \( 9^x - 3^{x+2} \) for all positive integer values of \( x \) is:
\[ y^2 - 9y \]