(D) $$ 9^{x}+3^{2 x .} $$ is $$ y^{x} $$

Question

UpStudy AI · Accepted Answer

To express $$ 9^{x} + 3^{2x} $$ in the form $$ y^{x} $$, let's simplify the expression step by step.

### Step 1: Simplify the Expression
First, recognize that $$ 9 $$ can be written as $$ 3^2 $$:

$$
9^{x} = (3^2)^{x} = 3^{2x}
$$

Now, substitute this back into the original expression:

$$
9^{x} + 3^{2x} = 3^{2x} + 3^{2x} = 2 \times 3^{2x}
$$

### Step 2: Express in Terms of $$ y^{x} $$
We want to express $$ 2 \times 3^{2x} $$ as $$ y^{x} $$:

$$
y^{x} = 2 \times 3^{2x}
$$

To solve for $$ y $$, take the $$ x $$-th root of both sides:

$$
y = \left(2 \times 3^{2x}\right)^{\frac{1}{x}} = 2^{\frac{1}{x}} \times 3^{2}
$$

Since $$ 3^2 = 9 $$, this simplifies to:

$$
y = 9 \times 2^{\frac{1}{x}}
$$

### Final Answer
$$
y = 9 \times 2^{\,1/x}
$$

**Alternatively**, using radical notation:

$$
y = 9 \cdot \sqrt[x]{2}
$$

This expresses $$ y $$ in terms of $$ x $$ such that $$ 9^{x} + 3^{2x} = y^{x} $$.

**Answer:**  
After simplifying, y is nine multiplied by 2¹⁄ˣ. Thus,

y = 9 · 2^(1/x)
$$ y = 9 \times 2^{\frac{1}{x}} $$

(D) \( 9^{x}+3^{2 x .} \) is \( y^{x} \)