Simplify the following $$ \div $$ $$ \begin{array}{ll} ext { 1) } \frac{2^{2 x}-2^{x}}{2^{x}-1} & ext { (2) } \frac{2^{2 x}-1}{2^{x}+1}\end{array} $$

Question

Simplify the following $$ \div $$ $$ \begin{array}{ll}	ext { 1) } \frac{2^{2 x}-2^{x}}{2^{x}-1} & 	ext { (2) } \frac{2^{2 x}-1}{2^{x}+1}\end{array} $$

UpStudy AI · Accepted Answer

Let's simplify each of the given expressions step by step.

### 1) $$\frac{2^{2x} - 2^{x}}{2^{x} - 1}$$

**Step 1:** Let $$ u = 2^{x} $$.

**Step 2:** Substitute $$ u $$ into the expression:
$$
\frac{u^2 - u}{u - 1}
$$

**Step 3:** Factor the numerator:
$$
\frac{u(u - 1)}{u - 1}
$$

**Step 4:** Cancel the common $$(u - 1)$$ terms (assuming $$ u \neq 1 $$):
$$
u = 2^{x}
$$

**Simplified Form:**
$$
2^{x}
$$

---

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

**Step 1:** Let $$ u = 2^{x} $$.

**Step 2:** Substitute $$ u $$ into the expression:
$$
\frac{u^2 - 1}{u + 1}
$$

**Step 3:** Factor the numerator using the difference of squares:
$$
\frac{(u - 1)(u + 1)}{u + 1}
$$

**Step 4:** Cancel the common $$(u + 1)$$ terms:
$$
u - 1 = 2^{x} - 1
$$

**Simplified Form:**
$$
2^{x} - 1
$$

---

**Final Answer:**

1) $$2^{x}$$

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

Simplify the following

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