Smpify: $$ \frac{4^{x x}-1}{4^{x}+1} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression:

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

let's explore possible factorizations and simplifications.

### Step 1: Express in Terms of Powers of 2
First, rewrite $$4^{x}$$ and $$4^{x^2}$$ in terms of powers of 2:

$$
4^{x} = (2^2)^{x} = 2^{2x}
$$
$$
4^{x^2} = (2^2)^{x^2} = 2^{2x^2}
$$

So the expression becomes:

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

### Step 2: Factor the Numerator
Notice that the numerator is a difference of squares:

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

Now the expression is:

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

### Step 3: Analyze for Further Simplification
At this point, we can look for common factors or further factorizations:

- The denominator $$2^{2x} + 1$$ doesn't directly factor in a way that would cancel with terms in the numerator unless specific conditions are met for $$x$$.
- Unless $$x$$ satisfies particular properties that make $$2^{2x} + 1$$ a factor of $$2^{x^2} \pm 1$$, no further simplification is possible.

### Conclusion
The expression does not simplify neatly into a more concise form without additional constraints or information about $$x$$. Therefore, the most simplified version of the expression is:

$$
\frac{4^{x^2} - 1}{4^{x} + 1} = \frac{2^{2x^2} - 1}{2^{2x} + 1} = \frac{(2^{x^2} - 1)(2^{x^2} + 1)}{2^{2x} + 1}
$$

Unless specific values or relationships for $$x$$ are provided, this is as simplified as the expression can get.
The expression simplifies to $$ \frac{4^{x^2} - 1}{4^{x} + 1} $$.

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