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a: \(a^2\cdot\sqrt{\dfrac{2}{3a}}=a^2\cdot\dfrac{\sqrt{2}}{\sqrt{3}\cdot\sqrt{a}}=\dfrac{a\sqrt{2}}{\sqrt{3}}=\dfrac{a\sqrt{6}}{3}\)

b: \(\dfrac{x-3}{x}\cdot\sqrt{\dfrac{x^3}{9-x^2}}\)

\(=\dfrac{x-3}{x}\cdot\dfrac{x\sqrt{x}}{\sqrt{x-3}\cdot\sqrt{x+3}}\)

\(=\dfrac{\sqrt{x}\cdot\sqrt{x-3}}{\sqrt{x+3}}\)

a) \(\left(\sqrt{28}-5\sqrt{35}+7\sqrt{112}\right)2\sqrt{7}=2\sqrt{196}-10\sqrt{245}+14\sqrt{784}\)

\(=28-10\sqrt{49.5}+392=420-70\sqrt{5}\)

b) \(\left(\sqrt{72}-3\sqrt{24}+5\sqrt{8}\right)\sqrt{2}+4\sqrt{27}=\sqrt{144}-3\sqrt{48}+5\sqrt{16}+4\sqrt{9.3}\)

\(=12-3\sqrt{16.3}+20+12\sqrt{3}=32-12\sqrt{3}+12\sqrt{3}=32\)

\(\sqrt{48\cdot45}=12\sqrt{15}\\ \sqrt{225\cdot17}=15\sqrt{17}\\ \sqrt{a^3b^7}=\left|ab^3\right|\sqrt{ab}=ab^3\sqrt{ab}\\ \sqrt{x^5\left(x-3\right)^2}=\left|x^2\left(x-3\right)\right|\sqrt{x}=x^2\left(x-3\right)\sqrt{x}\)

\(\sqrt{48\cdot45}=4\sqrt{3}\cdot3\sqrt{5}=12\sqrt{15}\)

\(\sqrt{225\cdot17}=15\sqrt{17}\)

a, \(\sqrt{54}=\sqrt{9.6}=3\sqrt{6}\)

b, \(\sqrt{108}=\sqrt{36.3}=6\sqrt{3}\)

c, \(0,1\sqrt{20000}=0,1\sqrt{2.10000}=10\sqrt{2}\)

d, \(-0,05\sqrt{28800}=-0,05\sqrt{288.100}=-0,05.10.\sqrt{144.2}\)

\(=-0,5.12\sqrt{2}=-6\sqrt{2}\)

e, \(\sqrt{7.63.a^2}=\sqrt{7.7.9.a^2}=21\left|a\right|\)

a) $\sqrt{54}=\sqrt{9.6}=\sqrt{9}.\sqrt{6}=3\sqrt{6}$.

b) $\sqrt{108}=\sqrt{36.3}=\sqrt{36}.\sqrt{3}=6\sqrt{3}$.

c) $0,1\sqrt{20000}=0,1\sqrt{2.10000}=0,1\sqrt{2}.\sqrt{10000}$

$=0,1.100\sqrt{2}=10\sqrt{2}$.

d) $-0,05\sqrt{28800}=-0,05\sqrt{288.100}$

$=-0,05\sqrt{\mathrm{2.144.100}}=-0,05.\sqrt{2}.\sqrt{144}.\sqrt{100}$

$=-0,\mathrm{05.12.10}\sqrt{2}$

$=-6\sqrt{2}$.
e) $\sqrt{\mathrm{7.63.}{a}^2}=\sqrt{\mathrm{7.7.9.}{a}^2}=\sqrt{7^2}.\sqrt{9}.\sqrt{a^2}$

$=\mathrm{7.3.}|a|=21|a|$

a) \(\sqrt{54}=\sqrt{9}.\sqrt{6}=3\sqrt{6}\)

b) \(\sqrt{108}=\sqrt{9}.\sqrt{4}.\sqrt{3}=6\sqrt{3}\)

c) \(0,1\sqrt{20000}=0,1.\sqrt{2}.\sqrt{10000}=0,1.100.\sqrt{2}=10\sqrt{2}\)

d) \(-0,05\sqrt{28800}=-0,05\sqrt{400}.\sqrt{9}.\sqrt[]{4}.\sqrt{2}=-0,05.20.3.2.\sqrt{2}=-6\sqrt{2}\)

e) \(\sqrt{7.63.a^2}=21\sqrt{a^2}\left\{{}\begin{matrix}=-21a\\=21a\end{matrix}\right.\)

\(\sqrt{18b^3\cdot\left(1-2a\right)^2}\)

\(=3\sqrt{2}\cdot b\sqrt{b}\cdot\left|1-2a\right|\)

\(=3\sqrt{2}\left(2a-1\right)\cdot b\sqrt{b}\)

\(\sqrt{34+24\sqrt{2}}=\sqrt{18+2.3\sqrt{2}.4+16}\)

\(=\sqrt{\left(3\sqrt{2}+4\right)^2}=3\sqrt{2}+4\)

a) \(\sqrt{27x^2}\)

\(=\sqrt{3^2\cdot3x^2}\)

\(=\left|3x\right|\sqrt{3}\)

\(=3\left|x\right|\sqrt{3}\)

b) \(\sqrt{8xy^2}\)

\(=\sqrt{2^2\cdot2\cdot x\cdot y^2}\)

\(=\left|2y\right|\sqrt{2x}\)

\(=2\left|y\right|\sqrt{2x}\)

c) \(\sqrt{25x^3}\)

\(=\sqrt{5^2\cdot x^2\cdot x}\)

\(=\left|5x\right|\sqrt{x}\)

\(=5\left|x\right|\sqrt{x}\)

d) \(\sqrt{48xy^4}\)

\(=\sqrt{4^2\cdot3x\cdot\left(y^2\right)^2}\)

\(=\left|4y^2\right|\sqrt{3x}\)

\(=4y^2\sqrt{3x}\)

`a, sqrt(27x^2b) = sqrt(3^2. 3.x^2b) = 3|x|sqrt(3b)`.

`b, sqrt(8xy^2) =sqrt(2^2.2xy^2)= 2|y|sqrt(2x)`

`c, sqrt(25x^3d) = sqrt(5^2.x^2.x.d) = 5|x|sqrt(xd)`.

`d, sqrt(48xy^4) = sqrt(4^2.3 . xy^4) = 4y^2sqrt(3x)`.