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The value of $$\sqrt{2023\sqrt{2022\sqrt{2021 \times 2019 + 1} + 1} + 1}$$ is
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The value of $$\sqrt{2023\sqrt{2022\sqrt{2021 \times 2019 + 1} + 1} + 1}$$ is
The innermost quantity is $$2021 \times 2019 + 1$$ which can be written as $$(2020+1)(2020-1) + 1$$
Using the property that $$(a+b)(a-b) = a^2 - b^2$$ we get
$$(2020+1)(2020-1) + 1 = 2020^2 -1^2 +1=2020^2$$
Hence, the square root of the innermost quantity is $$2020$$. The next radical becomes $$\sqrt{2022 \times 2020 + 1} $$
We can again rewrite this using the same property mentioned above to get $$(2021+1)(2021-1) + 1 = 2021^2-1^2 +1 = 2021^2$$
$$\sqrt{2021^2} = 2021$$. Now we can again rewrite this in the same way to get
$$\sqrt{2023 \times 2021 + 1} = \sqrt{(2022+1)(2022-1) +1} = \sqrt{2022^2-1^2+1}=2022$$.
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