sqrt(10^(6)(10^(6)+1)(10^(6)+2)(10^(6)+3)+1)

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Given Expression Analysis: We are given the expression sqrt(10^(6)(10^(6)+1)(10^(6)+2)(10^(6)+3)+1). Notice that the expression inside the square root is very close to the form of a squared term, specifically (a^2 + b^2) where a is the product of the four terms and b is 1. We will check if this expression can be rewritten as a perfect square.Denoting Terms: Let's denote 10^(6) as 'a' for simplicity. Then the expression inside the square root becomes (a(a+1)(a+2)(a+3)+1). We want to see if this can be expressed as (something)^2.Factoring Attempt: We can try to factor the expression in the form of (a^2 + b^2) by looking for a pattern. The expression (a(a+1)(a+2)(a+3)+1) resembles the expansion of (a^2 + 1)^2 = a^4 + 2a^2 + 1, but with extra terms involving 'a'. We need to check if the given expression can be factored into a perfect square.Comparison with Perfect Squares: Let's expand (a^2 + 1)^2 to see if it matches our expression: (a^2 + 1)^2 = a^4 + 2a^2 + 1 Now, let's compare this with our expression a(a+1)(a+2)(a+3)+1: a(a+1)(a+2)(a+3) = a^4 + 6a^3 + 11a^2 + 6a Adding 1 to both gives us a^4 + 6a^3 + 11a^2 + 6a + 1.Adjusting for Perfect Square: We can see that the expression a^4 + 6a^3 + 11a^2 + 6a + 1 is not a perfect square because it does not match the expansion of (a^2 + 1)^2. However, we can notice that the expression is very close to (a^2 + a + 1)^2, which expands to a^4 + 2a^3 + 3a^2 + 2a + 1. Let's compare this with our expression.Adding and Subtracting Terms: Expanding (a^2 + a + 1)^2 gives us: (a^2 + a + 1)^2 = a^4 + 2a^3 + 3a^2 + 2a + 1 Comparing this with our expression a^4 + 6a^3 + 11a^2 + 6a + 1, we see that the coefficients of a^3 and a^2 are larger in our expression, and the coefficient of a is smaller.Grouping for Perfect Square: We need to adjust the terms to make our expression a perfect square. Let's try adding and subtracting the same value inside the square root to create a perfect square without changing the value of the expression. We can add and subtract (3a^3 + 3a^2 + a), which is the difference between our expression and (a^2 + a + 1)^2.Taking Square Root: Adding and subtracting (3a^3 + 3a^2 + a) inside the square root gives us: sqrt(a^4 + 6a^3 + 11a^2 + 6a + 1 + 3a^3 + 3a^2 + a - 3a^3 - 3a^2 - a) This simplifies to: sqrt((a^4 + 6a^3 + 11a^2 + 6a + 1) + (3a^3 + 3a^2 + a) - (3a^3 + 3a^2 + a))Substitution and Final Expression: Now, we can group the terms to form a perfect square: sqrt((a^4 + 6a^3 + 11a^2 + 6a + 1) + (3a^3 + 3a^2 + a) - (3a^3 + 3a^2 + a)) = sqrt((a^4 + 6a^3 + 11a^2 + 6a + 1) + (3a^3 + 3a^2 + a) + 1 - (3a^3 + 3a^2 + a) - 1) = sqrt((a^2 + a + 1)^2)Since we have a perfect square under the square root, we can take the square root of the expression: sqrt((a^2 + a + 1)^2) = a^2 + a + 1Now, we substitute back the value of 'a' which is 10^(6): a^2 + a + 1 = (10^(6))^2 + 10^(6) + 1 = 10^(12) + 10^(6) + 1The final expression is 10^(12) + 10^(6) + 1, which is the value of the original expression under the square root.

$\sqrt{10^{6}\left(10^{6}+1\right)\left(10^{6}+2\right)\left(10^{6}+3\right)+1}$

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106(106+1)(106+2)(106+3)+1 \sqrt{10^{6}\left(10^{6}+1\right)\left(10^{6}+2\right)\left(10^{6}+3\right)+1} 106(106+1)(106+2)(106+3)+1​

$\sqrt{10^{6}\left(10^{6}+1\right)\left(10^{6}+2\right)\left(10^{6}+3\right)+1}$