12) $$ \frac{2^{x}-5 \cdot \sqrt{2^{x}}+6}{\sqrt{2^{x}}-2} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{2^{x} - 5 \cdot \sqrt{2^{x}} + 6}{\sqrt{2^{x}} - 2}
$$

**Step 1: Substitute for Simplification**

Let $$ y = \sqrt{2^{x}} $$. This implies that $$ 2^{x} = y^2 $$.

Substituting into the original expression:

$$
\frac{y^2 - 5y + 6}{y - 2}
$$

**Step 2: Factor the Numerator**

Factor the quadratic in the numerator:

$$
y^2 - 5y + 6 = (y - 2)(y - 3)
$$

So the expression becomes:

$$
\frac{(y - 2)(y - 3)}{y - 2}
$$

**Step 3: Simplify**

Assuming $$ y \neq 2 $$ (to avoid division by zero), we can cancel out $$ (y - 2) $$ from the numerator and the denominator:

$$
y - 3
$$

**Step 4: Substitute Back**

Recall that $$ y = \sqrt{2^{x}} $$. Therefore:

$$
\sqrt{2^{x}} - 3
$$

**Final Simplified Form:**

$$
\sqrt{2^{x}} - 3 \quad \text{or equivalently} \quad 2^{\frac{x}{2}} - 3
$$

**Conclusion:**

The simplified form of the given expression is:

$$
\boxed{\sqrt{2^{\,x}}\, -\, 3}
$$
The simplified form of the expression is $$ \sqrt{2^{x}} - 3 $$.

12) \( \frac{2^{x}-5 \cdot \sqrt{2^{x}}+6}{\sqrt{2^{x}}-2} \)