Question : What is the simplified value of $(x^{32}+\frac{1}{x^{32}})(x^{8}+\frac{1}{x^{8}})(x-\frac{1}{x})(x^{16}+\frac{1}{x^{16}})(x+\frac{1}{x})(x^{4}+\frac{1}{x^{4}})?$
Option 1: $(x^{64}+\frac{1}{x^{64}})$
Option 2: $\frac{(x^{64}-\frac{1}{x^{64}})}{(x^2+\frac{1}{x^2})}$
Option 3: $\frac{(x^{64}-\frac{1}{x^{64}})}{(x+\frac{1}{x})}$
Option 4: $\frac{(x^{32}-\frac{1}{x^{32}})}{(x+\frac{1}{x})}$
Correct Answer: $\frac{(x^{64}-\frac{1}{x^{64}})}{(x^2+\frac{1}{x^2})}$
Solution : Given: $(x^{32}+\frac{1}{x^{32}})(x^{8}+\frac{1}{x^{8}})(x-\frac{1}{x})(x^{16}+\frac{1}{x^{16}})(x+\frac{1}{x})(x^{4}+\frac{1}{x^{4}})$ $=(x^{32}+\frac{1}{x^{32}})(x^{8}+\frac{1}{x^{8}})(x^2-\frac{1}{x^2})(x^{16}+\frac{1}{x^{16}})(x^{4}+\frac{1}{x^{4}})$ Multiplying it by $\frac{(x^2+\frac{1}{x^2})}{(x^2+\frac{1}{x^2})}$, we get, $= \frac{(x^2+\frac{1}{x^2})}{(x^2+\frac{1}{x^2})}×(x^{32}+\frac{1}{x^{32}})(x^{8}+\frac{1}{x^{8}})(x^2-\frac{1}{x^2})(x^{16}+\frac{1}{x^{16}})(x^{4}+\frac{1}{x^{4}})$ $= \frac{1}{(x^2+\frac{1}{x^2})}×(x^{32}+\frac{1}{x^{32}})(x^{8}+\frac{1}{x^{8}})(x^{16}+\frac{1}{x^{16}})(x^{4}+\frac{1}{x^{4}})(x^{4}-\frac{1}{x^{4}})$ $= \frac{1}{(x^2+\frac{1}{x^2})}×(x^{32}+\frac{1}{x^{32}})(x^{8}+\frac{1}{x^{8}})(x^{16}+\frac{1}{x^{16}})(x^{8}-\frac{1}{x^{8}})$ $= \frac{1}{(x^2+\frac{1}{x^2})}×(x^{32}+\frac{1}{x^{32}})(x^{16}+\frac{1}{x^{16}})(x^{16}-\frac{1}{x^{16}})$ $= \frac{1}{(x^2+\frac{1}{x^2})}×(x^{32}+\frac{1}{x^{32}})(x^{32}-\frac{1}{x^{32}})$ $= \frac{1}{(x^2+\frac{1}{x^2})}×(x^{64}-\frac{1}{x^{64}})$ $= \frac{(x^{64}-\frac{1}{x^{64}})}{(x^2+\frac{1}{x^2})}$ Hence, the correct answer is $\frac{(x^{64}-\frac{1}{x^{64}})}{(x^2+\frac{1}{x^2})}$.
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Question : What is the simplified value of $(x^{128}+1)(x^{32}+1)(x^{64}+1)(x^{16}+1)(x^{8}+1)(x^{4}+1)(x^{2}+1)(x+1)?$
Option 1: $x^{256}-1$
Option 2: $\frac{x^{128}-1}{x-1}$
Option 3: $\frac{x^{64}-1}{x-1}$
Option 4: $\frac{x^{256}-1}{x-1}$
Question : What is the simplified value of $(3+1)(3^{2}+1)(3^{4}+1)(3^{8}+1)(3^{16}+1)$?
Option 1: $\frac{(3^{32}-1)}{2}$
Option 2: $\frac{(3^{16}-1)}{2}$
Option 3: $\frac{(3^{64}-1)}{2}$
Option 4: $\frac{(3^{128}-1)}{2}$
Question : What is the simplified value of $(2+1)(2^{2}+1)(2^{4}+1)(2^{8}+1)$?
Option 1: $2^{8}-1$
Option 2: $2^{16}-1$
Option 3: $2^{32}-1$
Option 4: $2^{64}-1$
Question : If $\frac{x^8+1}{x^4}=14$, then the value of $\frac{x^{12}+1}{x^6}$ is:
Option 1: 16
Option 2: 14
Option 3: 52
Option 4: 64
Question : What is the simplified value of: $\frac{1}{8}\left\{\left(x+\frac{1}{y}\right)^2-\left(x-\frac{1}{y}\right)^2\right\}$
Option 1: $\frac{x}{y}$
Option 2: $\frac{2x}{y}$
Option 3: $\frac{x}{2y}$
Option 4: $\frac{4x}{y}$
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