Sure, let’s solve the given expression step-by-step:
We start with the expression:
[tex]\[\left(x^{2x} - 4\right)\left(x^{2x} + 4\right)\][/tex]
We recognize that the expression is in the form of a product of two binomials.
Recall the identity for the difference of squares:
[tex]\[(a - b)(a + b) = a^2 - b^2\][/tex]
In this case, [tex]\(a\)[/tex] is [tex]\(x^{2x}\)[/tex] and [tex]\(b\)[/tex] is 4. Applying the difference of squares identity:
[tex]\[ (x^{2x} - 4)(x^{2x} + 4) = (x^{2x})^2 - 4^2 \][/tex]
Now, we simplify each term:
1. [tex]\((x^{2x})^2\)[/tex]:
[tex]\[ (x^{2x})^2 = x^{2x \cdot 2} = x^{4x} \][/tex]
2. [tex]\(4^2\)[/tex]:
[tex]\[ 4^2 = 16 \][/tex]
Putting it all together, we have:
[tex]\[ x^{4x} - 16 \][/tex]
So the simplified expression is:
[tex]\[ \left(x^{2x} - 4\right)\left(x^{2x} + 4\right) = x^{4x} - 16 \][/tex]
This is the final result.
