b) $$ \frac{2^{2 x}-1}{2^{x}+1} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{2^{2x} - 1}{2^x + 1}
$$

**Step 1: Recognize the Structure**

Notice that $$2^{2x}$$ can be written as $$(2^x)^2$$. This allows us to see the numerator as a difference of squares:

$$
2^{2x} - 1 = (2^x)^2 - 1^2
$$

**Step 2: Factor the Numerator**

Using the difference of squares formula, $$a^2 - b^2 = (a - b)(a + b)$$, we can factor the numerator:

$$
(2^x)^2 - 1^2 = (2^x - 1)(2^x + 1)
$$

**Step 3: Simplify the Expression**

Substitute the factored form back into the original expression:

$$
\frac{(2^x - 1)(2^x + 1)}{2^x + 1}
$$

Cancel out the common factor $$(2^x + 1)$$ in the numerator and the denominator:

$$
\frac{(2^x - 1)\cancel{(2^x + 1)}}{\cancel{2^x + 1}} = 2^x - 1
$$

**Final Simplified Form:**

$$
2^x - 1
$$

**Conclusion:**

The simplified form of the expression $$\frac{2^{2x} - 1}{2^x + 1}$$ is:

$$
2^x - 1
$$
The simplified form of the expression is $$2^x - 1$$.

b) \( \frac{2^{2 x}-1}{2^{x}+1} \)