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

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.