1.3 Simplify: $$ \frac{2^{x}-2^{-x}}{4^{x}-1} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\frac{2^{x} - 2^{-x}}{4^{x} - 1}
$$

Let's follow the steps below:

1. **Rewrite the Denominator:**

Notice that $$ 4^{x} $$ can be expressed as $$ (2^2)^x = 2^{2x} $$. Therefore, the denominator becomes:

$$
   4^{x} - 1 = 2^{2x} - 1
   $$

2. **Simplify the Numerator:**

The numerator $$ 2^{x} - 2^{-x} $$ can be rewritten using a common denominator:

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

3. **Combine Numerator and Denominator:**

Substitute the simplified numerator and denominator into the original expression:

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

4. **Cancel Common Terms:**

Notice that $$ 2^{2x} - 1 $$ appears in both the numerator and the denominator, so they cancel out:

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

**Final Simplified Form:**

$$
2^{-x}
$$

Answer: After simplifying, the expression is equal to 2 raised to –x. Thus,

2⁻ˣ
The simplified form of the expression is $$ 2^{-x} $$.

1.3 Simplify: