$\boldsymbol{\sqrt{8100000000180000000001}}$ $\boldsymbol{=\sqrt{8100000000000000000000+2\times90000000000\times1+1}}$ $\boldsymbol{=\sqrt{(90000000000+1)^2}}$ $=\boxed{90000000001}$

Obviously, this would be easy with a calculator. But we shall never be ruled by machines.

To solve, we use the pattern

$(a+b)^2=a^2+2ab+b^2$ .

In this question, there are three numerical components to the square number (separated nicely by zeroes), corresponding to the three components of $(a+b)^2$ . W.l.o.g., take $a$ as the larger component of $a+b$ . So we reason as follows, and hope for the best:

• $a^2=81\times10^{20}\Rightarrow a=\pm9\times10^{10}$ .

Since our answer is positive, this largest component must also be positive. So:

• $a=9\times10^{10}$ .

Now:

• $b^2=1\Rightarrow b=\pm1$ .

Given these values:

• $2ab=\pm18\times10^{10}$ .

Since we see $+18\times10^{10}$ in our square number (and not $82$ and loads of $9$ s, as we would with $ab$ negative), take

• $a=9\times10^{10}$
• $b=1$

and just make sure:

$(90000000001)^2$

$=(9\times10^{10}+1)^2$

$=81\times10^{20}+2(9\times10^{10})+1$

$=8100000000180000000001$ ,

as it should be.

So $\sqrt{8100000000180000000001}=\boxed{90000000001\,}$ .

$\begin{aligned} \sqrt{8100000000180000000001} & = \sqrt{81 \times 10^{20}+18\times 10^{10} + 1} \\ & = \sqrt{(9\times 10^{10})^2 + 2(9 \times 10^{10})(1) + 1^2} \\ & = \sqrt{(9\times 10^{10}+1)^2} \\ & = 9\times 10^{10}+1 \\ & = \boxed{90000000001} \end{aligned}$

(9^10 + 1)^2 = 90000000001^2 Therefore Root of 8100000000180000000001 = 90000000001

99=81 9+9=18 so we will use " a^2 + 2ab + b^2 " , a ^2 =( 9 * 10^ 10 )^2 b^2 = 1 2ab = 2(910^101) so, by applying on the pattern : 8110^20 + 1810^10 + 1 = the number above by sqrt both the number and the pattern that gives us (9*10^10 + 1) 90000000001

9 9=81 9+9=18 so we will use " a^2 + 2ab + b^2 " , a ^2 =( 9 * 10^ 10 )^2 b^2 = 1 2ab = 2 (9 10^10 1) so, by applying on the pattern : 81 10^20 + 18 10^10 + 1 = the number above by sqrt both the number and the pattern that gives us

(9*10^10 + 1)

90000000001