What is the simplified value of $\frac{1}{\sqrt{17}-\sqrt{16}}-\frac{1}{\sqrt{16}-\sqrt{15}}+\frac{1}{\sqrt{15}-\sqrt{14}}-\frac{1}{\sqrt{14}-\sqrt{13}}+\frac{1}{\sqrt{13}-\sqrt{12}}$?

Let's rationalize each term individually using the identity $\frac{1}{\sqrt{a}-\sqrt{b}}=\sqrt{a}+\sqrt{b}$ when $a-b=1$:$\frac{1}{\sqrt{17}-\sqrt{16}}=\sqrt{17}+\sqrt{16}$$\frac{1}{\sqrt{16}-\sqrt{15}}=\sqrt{16}+\sqrt{15}$$\frac{1}{\sqrt{15}-\sqrt{14}}=\sqrt{15}+\sqrt{14}$$\frac{1}{\sqrt{14}-\sqrt{13}}=\sqrt{14}+\sqrt{13}$$\frac{1}{\sqrt{13}-\sqrt{12}}=\sqrt{13}+\sqrt{12}$ Now substitute these values back into the expression:
$$E=(\sqrt{17}+\sqrt{16})-(\sqrt{16}+\sqrt{15})+(\sqrt{15}+\sqrt{14})-(\sqrt{14}+\sqrt{13})+(\sqrt{13}+\sqrt{12})$$
Open the brackets and simplify:
$$E=\sqrt{17}+\sqrt{16}-\sqrt{16}-\sqrt{15}+\sqrt{15}+\sqrt{14}-\sqrt{14}-\sqrt{13}+\sqrt{13}+\sqrt{12}$$
Most terms cancel out each other consecutively:
$$E=\sqrt{17}+\sqrt{12}$$