Evaluate
\sqrt{10}\approx 3.16227766
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\left(\sqrt{2}\right)^{2}+2\sqrt{2}\sqrt{5}+\left(\sqrt{5}\right)^{2}-\left(2+\sqrt{10}\right)^{2}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{2}+\sqrt{5}\right)^{2}.
2+2\sqrt{2}\sqrt{5}+\left(\sqrt{5}\right)^{2}-\left(2+\sqrt{10}\right)^{2}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)
The square of \sqrt{2} is 2.
2+2\sqrt{10}+\left(\sqrt{5}\right)^{2}-\left(2+\sqrt{10}\right)^{2}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)
To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.
2+2\sqrt{10}+5-\left(2+\sqrt{10}\right)^{2}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)
The square of \sqrt{5} is 5.
7+2\sqrt{10}-\left(2+\sqrt{10}\right)^{2}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)
Add 2 and 5 to get 7.
7+2\sqrt{10}-\left(4+4\sqrt{10}+\left(\sqrt{10}\right)^{2}\right)+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2+\sqrt{10}\right)^{2}.
7+2\sqrt{10}-\left(4+4\sqrt{10}+10\right)+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)
The square of \sqrt{10} is 10.
7+2\sqrt{10}-\left(14+4\sqrt{10}\right)+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)
Add 4 and 10 to get 14.
7+2\sqrt{10}-14-4\sqrt{10}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)
To find the opposite of 14+4\sqrt{10}, find the opposite of each term.
-7+2\sqrt{10}-4\sqrt{10}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)
Subtract 14 from 7 to get -7.
-7-2\sqrt{10}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)
Combine 2\sqrt{10} and -4\sqrt{10} to get -2\sqrt{10}.
-7-2\sqrt{10}+3\sqrt{10}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)
Factor 90=3^{2}\times 10. Rewrite the square root of the product \sqrt{3^{2}\times 10} as the product of square roots \sqrt{3^{2}}\sqrt{10}. Take the square root of 3^{2}.
-7+\sqrt{10}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)
Combine -2\sqrt{10} and 3\sqrt{10} to get \sqrt{10}.
-7+\sqrt{10}+\left(2\sqrt{2}\right)^{2}-1
Consider \left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.
-7+\sqrt{10}+2^{2}\left(\sqrt{2}\right)^{2}-1
Expand \left(2\sqrt{2}\right)^{2}.
-7+\sqrt{10}+4\left(\sqrt{2}\right)^{2}-1
Calculate 2 to the power of 2 and get 4.
-7+\sqrt{10}+4\times 2-1
The square of \sqrt{2} is 2.
-7+\sqrt{10}+8-1
Multiply 4 and 2 to get 8.
-7+\sqrt{10}+7
Subtract 1 from 8 to get 7.
\sqrt{10}
Add -7 and 7 to get 0.
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