What is sqrt(\sqrt{n(n+1)(n+2)(n+3)+1)-n} ?

The solution to sqrt(\sqrt{n(n+1)(n+2)(n+3)+1)-n} is sqrt(\sqrt{n^4+6n^3+11n^2+6n+1)-n}

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simplify sqrt(\sqrt{n(n+1)(n+2)(n+3)+1)-n}

simplify

n (n + 1) (n + 2) (n + 3) + 1 - n

n^{4} + 6 n^{3} + 11 n^{2} + 6 n + 1 - n

Solution steps

n (n + 1) (n + 2) (n + 3) + 1 - n

Apply FOIL method

(n + 1) (n + 2) : n^{2} + 3 n + 2

= n (n^{2} + 3 n + 2) (n + 3) + 1 - n

Expand

n (n + 3) : n^{2} + 3 n

= (n^{2} + 3 n) (n^{2} + 3 n + 2) + 1 - n

Distribute parentheses

= n^{2} n^{2} + n^{2} \cdot 3 n + n^{2} \cdot 2 + 3 n n^{2} + 3 n \cdot 3 n + 3 n \cdot 2 + 1 - n

Simplify

n^{2} n^{2} + n^{2} \cdot 3 n + n^{2} \cdot 2 + 3 n n^{2} + 3 n \cdot 3 n + 3 n \cdot 2 : n^{4} + 6 n^{3} + 11 n^{2} + 6 n

= n^{4} + 6 n^{3} + 11 n^{2} + 6 n + 1 - n

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What is sqrt(\sqrt{n(n+1)(n+2)(n+3)+1)-n} ?
The solution to sqrt(\sqrt{n(n+1)(n+2)(n+3)+1)-n} is sqrt(\sqrt{n^4+6n^3+11n^2+6n+1)-n}