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Simplify the expression\newline(log⁡np)4+(log⁡pn)4+2+2−log⁡np−log⁡pn\sqrt{\sqrt{(\log_{n}p)^{4}+(\log_{p}n)^{4}+2}+2}-\log_{n}p-\log_{p}n

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Full solution

Q. Simplify the expression\newline(log⁡np)4+(log⁡pn)4+2+2−log⁡np−log⁡pn\sqrt{\sqrt{(\log_{n}p)^{4}+(\log_{p}n)^{4}+2}+2}-\log_{n}p-\log_{p}n
  1. Simplify Innermost Square Root: First, let's simplify the inside of the innermost square root: (log⁡np)4+(log⁡pn)4+2(\log_{n}p)^{4}+(\log_{p}n)^{4}+2. Notice that (log⁡np)(\log_{n}p) and (log⁡pn)(\log_{p}n) are inverses of each other, so (log⁡np)⋅(log⁡pn)=1(\log_{n}p) \cdot (\log_{p}n) = 1.
  2. Rewrite Using Property: Now, let's rewrite the expression using the property that (log⁡np)2⋅(log⁡pn)2=12=1(\log_{n}p)^{2} \cdot (\log_{p}n)^{2} = 1^{2} = 1. So, (log⁡np)4+(log⁡pn)4+2(\log_{n}p)^{4}+(\log_{p}n)^{4}+2 can be written as ((log⁡np)2)2+((log⁡pn)2)2+2⋅1⋅1((\log_{n}p)^{2})^{2} + ((\log_{p}n)^{2})^{2} + 2 \cdot 1 \cdot 1. This looks like the formula a2+b2+2aba^{2} + b^{2} + 2ab, which is (a+b)2(a+b)^{2}.
  3. Simplify Expression: So, we have (log⁡np)2+(log⁡pn)22+2\sqrt{\sqrt{(\log_{n}p)^{2} + (\log_{p}n)^{2}}^{2} + 2}. This simplifies to (log⁡np)2+(log⁡pn)2+2\sqrt{(\log_{n}p)^{2} + (\log_{p}n)^{2} + 2}.
  4. Incorrect Simplification: Now, we simplify the outer square root: (log⁡np)2+(log⁡pn)2+2\sqrt{(\log_{n}p)^{2} + (\log_{p}n)^{2} + 2} = (log⁡np)2\sqrt{(\log_{n}p)^2} + (log⁡pn)2\sqrt{(\log_{p}n)^2} + 2\sqrt{2}. This is incorrect because we cannot split the square root over addition like this. The correct step should have been to recognize that the expression under the square root is already a perfect square, so we should take the square root of the entire expression.

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