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

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

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

$-0,1\sqrt{100z^{10}}=-0,1\sqrt{(10z^5{)}^2}=-0,1\ast |10z^5|=-0,1\ast 10z^5=-z^5$, 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

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

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

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