Rewrite $\sec \left(\sin ^{-1} 4 w\right)$ as an algebraic expression in $w$.

Step 1 ：Rewrite \(\sec \left(\sin ^{-1} 4 w\right)\) as an algebraic expression in \(w\).

Step 2 ：The secant function is the reciprocal of the cosine function. So, \(\sec(x) = \frac{1}{\cos(x)}\).

Step 3 ：The inverse sine function, \(\sin^{-1}(x)\), gives the angle whose sine is \(x\). So, \(\sin^{-1}(4w)\) gives the angle whose sine is \(4w\).

Step 4 ：We can use the Pythagorean identity, \(\sin^2(x) + \cos^2(x) = 1\), to express cosine in terms of sine: \(\cos(x) = \sqrt{1 - \sin^2(x)}\).

Step 5 ：So, \(\sec(\sin^{-1}(4w)) = \frac{1}{\cos(\sin^{-1}(4w))} = \frac{1}{\sqrt{1 - \sin^2(\sin^{-1}(4w))}} = \frac{1}{\sqrt{1 - (4w)^2}}\).

Step 6 ：However, we need to consider the domain of the inverse sine function, which is \(-1 \leq x \leq 1\). So, \(-1 \leq 4w \leq 1\), which means \(-1/4 \leq w \leq 1/4\).

Step 7 ：Final Answer: \(\boxed{\frac{1}{\sqrt{1 - 16w^2}}}\), for \(-1/4 \leq w \leq 1/4\).