$\sqrt{9a^{16}}=\sqrt{(3a^8{)}^2}=|3a^8|=3a^8$

$\sqrt{0,81d^6}=\sqrt{(0,9d^3{)}^2}=|0,9d^3|=0,9d^3$, если d ≥ 0

$-<!--\; -\; -->5\sqrt{4x^2}=-<!--\; -\; -->5\sqrt{(2x{)}^2}=-<!--\; -\; -->5\ast <!--\; \ast \; -->|2x|=-<!--\; -\; -->5\ast <!--\; \ast \; -->(-<!--\; -\; -->2x)=10x$, если x ≤ 0

$-<!--\; -\; -->0,1\sqrt{100z^{10}}=-<!--\; -\; -->0,1\sqrt{(10z^5{)}^2}=-<!--\; -\; -->0,1\ast <!--\; \ast \; -->|10z^5|=-<!--\; -\; -->0,1\ast <!--\; \ast \; -->10z^5=-<!--\; -\; -->z^5$, если z ≥ 0

$\sqrt{p^6{q}^8}=\sqrt{(p^3{q}^4{)}^2}=|p^3{q}^4|=p^3{q}^4$, если p ≥ 0

$\sqrt{25m^{34}{n}^{38}}=\sqrt{(5m^{17}{n}^{19}{)}^2}=|5m^{17}{n}^{19}|=5\ast <!--\; \ast \; -->(-<!--\; -\; -->m^{1}7)\ast <!--\; \ast \; -->(-<!--\; -\; -->n^{1}9)=5m^{17}{n}^{19}$, если m ≤ 0, n ≤ 0

$a{b}^2\sqrt{a^4{b}^{18}{c}^{22}}=a{b}^2\sqrt{(a^2{b}^9{c}^{11}{)}^2}=a{b}^2\ast <!--\; \ast \; -->|a^2{b}^9{c}^{11}|=a{b}^2\ast <!--\; \ast \; -->a^2{b}^9\ast <!--\; \ast \; -->(-<!--\; -\; -->c^{11})=-<!--\; -\; -->a^3{b}^{11}{c}^{11}$, если b ≥ 0, c ≤ 0

$-<!--\; -\; -->\frac{8m^3{p}^4}{k^2}\sqrt{\frac{625k^{30}{p}^{40}}{144m^6}}=-<!--\; -\; -->\frac{8m^3{p}^4}{k^2}\ast <!--\; \ast \; -->\frac{\sqrt{625k^{30}{p}^{40}}}{\sqrt{144m^6}}=-<!--\; -\; -->\frac{8m^3{p}^4}{k^2}\ast <!--\; \ast \; -->\frac{\sqrt{(25k^{15}{p}^{20}{)}^2}}{\sqrt{(12m^3{)}^2}}=-<!--\; -\; -->\frac{8m^3{p}^4}{k^2}\ast <!--\; \ast \; -->\frac{|25k^{15}{p}^{20}|}{|12m^3|}=-<!--\; -\; -->\frac{8m^3{p}^4}{k^2}\ast <!--\; \ast \; -->\frac{25k^{15}{p}^{20}}{-<!--\; -\; -->12m^3}=\frac{2p^4}{1}\ast <!--\; \ast \; -->\frac{25k^{13}{p}^{20}}{3}=\frac{50k^{13}{p}^{24}}{3}$, если m <0, k > 0