$$ \frac{\operatorname{li}^{ ext {2xv2 }}-3^{x-1} \cdot 3^{x+1}}{9 \cdot 3^{2 x-2}} $$

Question

$$ \frac{\operatorname{li}^{	ext {2xv2 }}-3^{x-1} \cdot 3^{x+1}}{9 \cdot 3^{2 x-2}} $$

UpStudy AI · Accepted Answer

Let's simplify the given expression step by step:

$$
\frac{\operatorname{li}^{\text{2xv2}} - 3^{x-1} \cdot 3^{x+1}}{9 \cdot 3^{2x-2}}
$$

**Step 1: Simplify the Exponents with the Same Base**

Notice that $$3^{x-1} \cdot 3^{x+1}$$ can be simplified using the properties of exponents:

$$
3^{x-1} \cdot 3^{x+1} = 3^{(x-1) + (x+1)} = 3^{2x}
$$

**Step 2: Simplify the Denominator**

The denominator is:

$$
9 \cdot 3^{2x-2}
$$

Since $$9 = 3^2$$, we can rewrite the denominator as:

$$
3^2 \cdot 3^{2x-2} = 3^{2 + 2x - 2} = 3^{2x}
$$

**Step 3: Rewrite the Entire Expression**

Substituting the simplified numerator and denominator back into the original expression:

$$
\frac{\operatorname{li}^{\text{2xv2}} - 3^{2x}}{3^{2x}}
$$

This can be separated into two terms:

$$
\frac{\operatorname{li}^{\text{2xv2}}}{3^{2x}} - \frac{3^{2x}}{3^{2x}} = \frac{\operatorname{li}^{\text{2xv2}}}{3^{2x}} - 1
$$

**Final Simplified Form:**

$$
\frac{\operatorname{li}^{\text{2xv2}}}{3^{2x}} - 1
$$

**Notes:**

1. **Understanding $$\operatorname{li}$$:** The operator $$\operatorname{li}$$ typically refers to the logarithmic integral function, defined as:

$$
   \operatorname{li}(x) = \int_0^x \frac{dt}{\ln t}
   $$

However, in your expression, $$\operatorname{li}$$ is raised to the power of $$\text{2xv2}$$. Ensure that this is intentional and that $$\text{2xv2}$$ correctly represents the exponent you intend to use.

2. **Interpreting $$\text{2xv2}$$:** If "v2" was meant to represent $$\sqrt{2}$$, then $$\text{2xv2} = 2x\sqrt{2}$$. Ensure that the exponent is correctly notated to avoid confusion.

3. **Final Expression:**

$$
   \frac{\operatorname{li}^{2x\sqrt{2}}}{3^{2x}} - 1
   $$

If $$\text{v2}$$ indeed represents $$\sqrt{2}$$, the expression above reflects that interpretation.

If you intended something different with the notation, please provide further clarification!
$$ \frac{\operatorname{li}^{2x\sqrt{2}}}{3^{2x}} - 1 $$

\( \frac{\operatorname{li}^{\text {2xv2 }}-3^{x-1} \cdot 3^{x+1}}{9 \cdot 3^{2 x-2}} \)

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