Bài 5:
a: \(\sqrt{148^2-48^2}\)
\(=\sqrt{\left(148-48\right)\left(148+48\right)}\)
\(=\sqrt{100\cdot196}=10\cdot14=140\)
b: \(\frac{3-\sqrt7}{3+\sqrt7}-\frac{3+\sqrt7}{3-\sqrt7}\)
\(=\frac{\left(3-\sqrt7\right)^2-\left(3+\sqrt7\right)^2}{\left(3-\sqrt7\right)\left(3+\sqrt7\right)}\)
\(=\frac{16-6\sqrt7-16-6\sqrt7}{9-7}=-\frac{12\sqrt7}{2}=-6\sqrt7\)
c: \(\sqrt{13+6\cdot\sqrt{4+\sqrt{9-4\sqrt2}}}\)
\(=\sqrt{13+6\cdot\sqrt{4+\sqrt{\left(2\sqrt2-1\right)^2}}}\)
\(=\sqrt{13+6\cdot\sqrt{4+\left(2\sqrt2-1\right)}}\)
\(=\sqrt{13+6\cdot\sqrt{3+2\sqrt2}}=\sqrt{13+6\left(\sqrt2+1\right)}=\sqrt{19+6\sqrt2}=3\sqrt2+1\)
d: \(\left(4+\sqrt{15}\right)\left(\sqrt{10}-\sqrt6\right)\cdot\sqrt{4-\sqrt{15}}\)
\(=\left(4+\sqrt{15}\right)\left(\sqrt5-\sqrt3\right)\cdot\sqrt{8-2\sqrt{15}}\)
\(=\left(4+\sqrt{15}\right)\left(\sqrt5-\sqrt3\right)\cdot\left(\sqrt5-\sqrt3\right)\)
\(=\left(4+\sqrt{15}\right)\left(8-2\sqrt{15}\right)=2\left(4+\sqrt{15}\right)\left(4-\sqrt{15}\right)=2\)
e: \(\left(3+\sqrt5\right)\left(\sqrt{10}-\sqrt2\right)\cdot\sqrt{3-\sqrt5}\)
\(=\left(3+\sqrt5\right)\left(\sqrt5-1\right)\cdot\sqrt{6-2\sqrt5}\)
\(=\left(3+\sqrt5\right)\left(\sqrt5-1\right)\cdot\sqrt{\left(\sqrt5-1\right)^2}=\left(3+\sqrt5\right)\left(\sqrt5-1\right)\left(\sqrt5-1\right)\)
\(=\left(3+\sqrt5\right)\left(6-2\sqrt5\right)=2\left(3+\sqrt5\right)\left(3-\sqrt5\right)=2\left(9-5\right)=2\cdot4=8\)
f: \(\left(\sqrt3-1\right)\cdot\sqrt{2\cdot\sqrt{19+8\sqrt3}-4}\)
\(=\left(\sqrt3-1\right)\cdot\sqrt{2\cdot\left(4+\sqrt3\right)-4}\)
\(=\left(\sqrt3-1\right)\cdot\sqrt{8+2\sqrt3-4}\)
\(=\left(\sqrt3-1\right)\cdot\sqrt{4+2\sqrt3}=\left(\sqrt3-1\right)\left(\sqrt3+1\right)=3-1=2\)
g: \(\sqrt{47+\sqrt5}\cdot\sqrt{7-\sqrt{2+\sqrt5}}\cdot\sqrt{7+\sqrt{2+\sqrt5}}\)
\(=\sqrt{47+\sqrt5}\cdot\sqrt{49-\left(2+\sqrt5\right)}\)
\(=\sqrt{47+\sqrt5}\cdot\sqrt{47-\sqrt5}=\sqrt{47^2-5}=2\sqrt{551}\)
Bài 3:
a: \(\sqrt{\frac{4}{81}}=\sqrt{\frac{2^2}{9^2}}=\frac29\)
b: \(\sqrt{\frac{9}{25}}=\sqrt{\left(\frac35\right)^2}=\frac35\)
c: \(\sqrt{\frac{\left(-2\right)^4}{5^6}}=\sqrt{\frac{4^2}{\left(5^3\right)^2}}=\frac{4}{5^3}=\frac{4}{125}\)
d: \(\sqrt{\frac{4a^2b^6}{9b^4}}=\sqrt{\frac49a^2b^2}=\frac23\cdot\left|ab\right|\)
