7. for what value of x is the expression $(x^{2}+4)^{2}(x^{3}+8)^{-3}(x^{4}+16)^{4}$ undefined?\na. -8\nb…

Explanation:

Step1: Recall undefined condition

A rational - function is undefined when the denominator is zero. Here, we have a factor ((x^{3}+8)^{-3}=\frac{1}{(x^{3}+8)^{3}}). The expression will be undefined when (x^{3}+8 = 0).

Step2: Solve the equation (x^{3}+8 = 0)

We can rewrite (x^{3}+8) as (x^{3}+2^{3}). Using the sum - of - cubes formula (a^{3}+b^{3}=(a + b)(a^{2}-ab + b^{2})), where (a=x) and (b = 2), we have ((x + 2)(x^{2}-2x + 4)=0). The quadratic factor (x^{2}-2x + 4) has discriminant (\Delta=b^{2}-4ac=(-2)^{2}-4\times1\times4=4 - 16=-12<0), so its roots are complex. Setting (x + 2=0), we get (x=-2). Also, note that (x^{2}+4>0) for all real (x) (since (x^{2}\geq0) for all real (x), then (x^{2}+4\geq4>0)) and (x^{4}+16>0) for all real (x) (since (x^{4}\geq0) for all real (x), then (x^{4}+16\geq16>0)).

Answer:

C. -2