a) \(\left(4\sqrt{2}+\sqrt{30}\right)\left(\sqrt{5}-\sqrt{3}\right).\sqrt{4-\sqrt{15}}\)

\(=\left(4\sqrt{10}-4\sqrt{6}+\sqrt{150}-\sqrt{90}\right).\sqrt{\dfrac{8-2\sqrt{15}}{2}}\)

\(=\left(4\sqrt{10}-4\sqrt{6}+\sqrt{25.6}-\sqrt{9.10}\right).\sqrt{\dfrac{\left(\sqrt{5}\right)^2-2\sqrt{5}.\sqrt{3}+\left(\sqrt{3}\right)^2}{2}}\)

\(=\left(4\sqrt{10}-4\sqrt{6}+5\sqrt{6}-3\sqrt{10}\right).\sqrt{\dfrac{\left(\sqrt{5}-\sqrt{3}\right)^2}{2}}\)

\(=\left(\sqrt{10}+\sqrt{6}\right).\dfrac{\left|\sqrt{5}-\sqrt{3}\right|}{\sqrt{2}}=\sqrt{2}.\left(\sqrt{5}+\sqrt{3}\right).\dfrac{\sqrt{5}-\sqrt{3}}{\sqrt{2}}\)

\(=\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)=2\)

a) Ta có: \(\left(4\sqrt{2}+\sqrt{30}\right)\left(\sqrt{5}-\sqrt{3}\right)\cdot\sqrt{4-\sqrt{15}}\)

\(=\sqrt{8-2\sqrt{15}}\cdot\left(4+\sqrt{15}\right)\left(\sqrt{5}-\sqrt{3}\right)\)

\(=\left(\sqrt{5}-\sqrt{3}\right)^2\cdot\left(4+\sqrt{15}\right)\)

\(=\left(8-2\sqrt{15}\right)\left(4+\sqrt{15}\right)\)

\(=32+8\sqrt{15}-8\sqrt{15}-30\)

=2

\(\left(\sqrt{A}+\sqrt{B}\right)^2\)\(=A+B+2\sqrt{AB}\)

\(\left(\sqrt{A}-\sqrt{B}\right)^2\)\(=A-B+2\sqrt{AB}\)

a, \(VT=\dfrac{\sqrt{ab}\left(\sqrt{a}+\sqrt{b}\right)}{\sqrt{ab}}.\left(\sqrt{a}-\sqrt{b}\right)=a-b=VP\) đpcm

b,\(VT=1-\dfrac{a-\sqrt{a}}{\sqrt{a}-1}+\dfrac{a+\sqrt{a}}{\sqrt{a}+1}-\dfrac{a^2-a}{a-1}=1-\sqrt{a}+\sqrt{a}-a=1-a=VP\) đpcm

B1:

1. \(\sqrt{12.5}\cdot\sqrt{0.2}\cdot\sqrt{0.1}\) \(=\sqrt{12.5\cdot0.2\cdot0.1}\) \(=\sqrt{0.25}=0.5\)

2.\(\sqrt{48.4}\cdot\sqrt{5}\cdot\sqrt{0.5}\) = \(\sqrt{48.4\cdot5\cdot0.5}\) =\(\sqrt{121}=11\)

B2:

a, \(\left(\sqrt{7}+\sqrt{3}\right)^2=7+2\cdot\sqrt{7}\cdot\sqrt{3}+3=7+2\cdot\sqrt{21}+3\)\(=10+2\sqrt{21}\)

b,\(\left(\sqrt{11}-\sqrt{5}\right)^2=11-2\sqrt{55}+5=16-2\sqrt{55}\)

c,\(\left(\sqrt{x}+\sqrt{y}\right) ^2=x+2\sqrt{xy}+y\)

d.\(\left(\sqrt{13}+\sqrt{7}\right)^2=13+2\sqrt{7}+7=20+2\sqrt{7}\)

e,\(\left(\sqrt{a}-\sqrt{b}\right)^2=a-2\sqrt{ab}+b\)

f,\(\left(\sqrt{3}-1\right)^2=3-2\sqrt{3}+1=4-2\sqrt{3}\)

a. Sửa đề: \(\left(3+\sqrt{5}\right)\left(\sqrt{10}-\sqrt{2}\right)\sqrt{3-\sqrt{5}}=8\)

biến đổi vế trái :
ta có :\(\left(3+\sqrt{5}\right)\left(\sqrt{10}+\sqrt{2}\right)\sqrt{3-\sqrt{5}}\)

=\(\sqrt{3+\sqrt{5}}.\sqrt{3+\sqrt{5}}.\left(\sqrt{10}-\sqrt{2}\right).\sqrt{3-\sqrt{5}}\)

=\(\sqrt{3^2-\left(\sqrt{5}\right)^2}.\sqrt{3+\sqrt{5}}.\left(\sqrt{10}-\sqrt{2}\right)\)

=2(\(\sqrt{30+10\sqrt{5}}-\sqrt{6+2\sqrt{5}}\))

=2(\(\sqrt{5}+5-\sqrt{5}-1\))

=2.4=8=VP
=> đpcm

b. Đặt vế trái là A
ta có \(A^2=\sqrt{2}+1-2\sqrt{\left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right)}+\sqrt{2}-1\)

=\(2\sqrt{2}-2\)

=2\(\left(\sqrt{2}-1\right)\)

=> A=\(\sqrt{2\left(\sqrt{2}-1\right)}\)

vậy VT=VP =>đpcm

Bài 1:

a) Ta có: \(\sqrt{\left(23-15\sqrt{3}\right)^2}\)

\(=\left|23-15\sqrt{3}\right|\)

\(=\left|\sqrt{529}-\sqrt{675}\right|\)

\(=\sqrt{675}-\sqrt{529}\)

\(=15\sqrt{3}-23\)

b) Ta có: \(\sqrt{\left(2-2\sqrt{3}\right)^2}\)

\(=\left|2-2\sqrt{3}\right|\)

\(=2\sqrt{3}-2\)

c) Ta có: \(\sqrt{\left(15-4\sqrt{3}\right)^2}\)

\(=\left|15-4\sqrt{3}\right|\)

\(=15-4\sqrt{3}\)

d) Ta có: \(\sqrt{\left(16-6\sqrt{7}\right)^2}\)

\(=\left|16-6\sqrt{7}\right|\)

\(=\left|\sqrt{256}-\sqrt{252}\right|\)

\(=16-6\sqrt{7}\)

f) Ta có: \(\sqrt{\left(22-8\sqrt{3}\right)^2}\)

\(=\left|22-8\sqrt{3}\right|\)

\(=\left|\sqrt{484}-\sqrt{192}\right|\)

\(=22-8\sqrt{3}\)

g) Ta có: \(\sqrt{\left(9-4\sqrt{2}\right)^2}\)

\(=\left|9-4\sqrt{2}\right|\)

\(=9-4\sqrt{2}\)

h) Ta có: \(\sqrt{\left(13-4\sqrt{3}\right)^2}\)

\(=\left|13-4\sqrt{3}\right|\)

\(=13-4\sqrt{3}\)

i) Ta có: \(\sqrt{\left(7-3\sqrt{3}\right)^2}\)

\(=\left|7-3\sqrt{3}\right|\)

\(=7-3\sqrt{3}\)