a.	$\begin{align}&\sqrt {20}\\[5pt] &=\sqrt {4 \times 5}\\[5pt] &=\sqrt {4} \times \sqrt {5}\\[5pt] &=2 \times \sqrt {5}\\[5pt] &=\underline{\underline{2 \sqrt {5}}}\end{align}$	b.	$\begin{align}&\sqrt {48}\\[5pt] &=\sqrt {16 \times 3}\\[5pt] &=\sqrt {16} \times \sqrt {3}\\[5pt] &=4 \times \sqrt {3}\\[5pt] &=\underline{\underline{4 \sqrt {3}}}\end{align}$	c.	$\begin{align}&\sqrt {72}\\[5pt] &=\sqrt {36 \times 2}\\[5pt] &=\sqrt {36} \times \sqrt {2}\\[5pt] &=6 \times \sqrt {2}\\[5pt] &=\underline{\underline{6 \sqrt {2}}}\end{align}$
d.	$\begin{align}&\sqrt {28}\\[5pt] &=\sqrt {4 \times 7}\\[5pt] &=\sqrt {4} \times \sqrt {7}\\[5pt] &=2 \times \sqrt {7}\\[5pt] &=\underline{\underline{2 \sqrt {7}}}\end{align}$	e.	$\begin{align}&\sqrt {80}\\[5pt] &=\sqrt {16 \times 5}\\[5pt] &=\sqrt {16} \times \sqrt {5}\\[5pt] &=4 \times \sqrt {5}\\[5pt] &=\underline{\underline{4 \sqrt {5}}}\end{align}$	f.	$\begin{align}&\sqrt {45}\\[5pt] &=\sqrt {9 \times 5}\\[5pt] &=\sqrt {9} \times \sqrt {5}\\[5pt] &=3 \times \sqrt {5}\\[5pt] &=\underline{\underline{3 \sqrt {5}}}\end{align}$
g.	$\begin{align}&\sqrt {75}\\[5pt] &=\sqrt {25 \times 3}\\[5pt] &=\sqrt {25} \times \sqrt {3}\\[5pt] &=5 \times \sqrt {3}\\[5pt] &=\underline{\underline{5 \sqrt {3}}}\end{align}$	h.	$\begin{align}&\sqrt {147}\\[5pt] &=\sqrt {49 \times 3}\\[5pt] &=\sqrt {49} \times \sqrt {3}\\[5pt] &=7 \times \sqrt {3}\\[5pt] &=\underline{\underline{7 \sqrt {3}}}\end{align}$

a.	$\begin{align}&2\sqrt {3}\\[5pt] &=\sqrt {4} \times \sqrt {3}\\[5pt] &=\sqrt {4 \times 3}\\[5pt] &= \underline{\underline{\sqrt {12}}}\end{align}$	b.	$\begin{align}&2\sqrt {5}\\[5pt] &=\sqrt {4} \times \sqrt {5}\\[5pt] &=\sqrt {4 \times 5}\\[5pt] &= \underline{\underline{\sqrt {20}}}\end{align}$	c.	$\begin{align}&4\sqrt {7}\\[5pt] &=\sqrt {16} \times \sqrt {7}\\[5pt] &=\sqrt {16 \times 7}\\[5pt] &= \underline{\underline{\sqrt {112}}}\end{align}$
d.	$\begin{align}&5\sqrt {2}\\[5pt] &=\sqrt {25} \times \sqrt {2}\\[5pt] &=\sqrt {25 \times 2}\\[5pt] &= \underline{\underline{\sqrt {50}}}\end{align}$	e.	$\begin{align}&6\sqrt {11}\\[5pt] &=\sqrt {36} \times \sqrt {11}\\[5pt] &=\sqrt {36 \times 11}\\[5pt] &= \underline{\underline{\sqrt {396}}}\end{align}$

a.	$\begin{align}&\sqrt {2}+5\sqrt {2}-2\sqrt {2}\\[5pt] &=6\sqrt {2} - 2\sqrt {2}\\[5pt] &= \underline{\underline{4\sqrt {2}}}\end{align}$
b.	$\begin{align}&\sqrt {5}+2\sqrt {7}+2\sqrt {5}-3\sqrt {7}\\[5pt] &= \sqrt {5}+2\sqrt {5}+2\sqrt {7}-3\sqrt {7}\\[5pt] &= \underline{\underline{3\sqrt {5}-\sqrt {7}}}\end{align}$
c.	$\begin{align}&4\sqrt {3}+5\sqrt {2}+3\sqrt {5}-3\sqrt {2}+3\sqrt {5}-2\sqrt {3}\\[5pt] &= 4\sqrt {3}-2\sqrt {3}+5\sqrt {2}-3\sqrt {2}+3\sqrt {5}+3\sqrt {5}\\[5pt] &= \underline{\underline{2\sqrt {3}+2\sqrt {2}+6\sqrt {5}}}\end{align}$
d.	$\begin{align}&6\sqrt {11}+3\sqrt {7}-2\sqrt {11}-5\sqrt {7}+4\sqrt {7}\\[5pt] &=6\sqrt {11}-2\sqrt {11}+3\sqrt {7}+4\sqrt {7}-5\sqrt {7}\\[5pt] &= \underline{\underline{4\sqrt {11}+2\sqrt {7}}}\end{align}$
e.	$\begin{align}&8\sqrt {3}+7\sqrt {7}-2\sqrt {3}+3\sqrt {7}-3\sqrt {7}\\[5pt] &=8\sqrt {3}-2\sqrt {3}+7\sqrt {7}+3\sqrt {7}-3\sqrt {7}\\[5pt] &= \underline{\underline{6\sqrt {3}+7\sqrt {7}}}\end{align}$

a.	$\begin{align}&\dfrac{2} {\sqrt{5}}\\[5pt] &= \dfrac{2 \times \sqrt{5}} {\sqrt{5} \times \sqrt{5}}\\[5pt] &= \underline{\underline{\dfrac{2 \sqrt{5}} {5}}}\end{align}$	b.	$\begin{align}&\dfrac{5} {\sqrt{3}}\\[5pt] &= \dfrac{5 \times \sqrt{3}} {\sqrt{3} \times \sqrt{3}}\\[5pt] &= \underline{\underline{\dfrac{5 \sqrt{3}} {3}}}\end{align}$	c.	$\begin{align}&\dfrac{5} {\sqrt{7}}\\[5pt] &= \dfrac{5 \times \sqrt{7}} {\sqrt{7} \times \sqrt{7}}\\[5pt] &= \underline{\underline{\dfrac{5 \sqrt{7}} {7}}}\end{align}$
d.	$\begin{align}&\dfrac{12} {2\sqrt{3}}\\[5pt] &= \dfrac{12 \times \sqrt{3}} {2\sqrt{3} \times \sqrt{3}}\\[5pt] &= \dfrac{12 \sqrt{3}} {2\times 3}\\[5pt] &= \dfrac{12 \sqrt{3}} {6}\\[5pt] &= \underline{\underline{2 \sqrt{3}}}\end{align}$	e.	$\begin{align}&\dfrac{27} {3\sqrt{2}}\\[5pt] &= \dfrac{27 \times \sqrt{2}} {3\sqrt{2} \times \sqrt{2}}\\[5pt] &= \dfrac{27 \sqrt{2}} {3\times 2}\\[5pt] &= \dfrac{27 \sqrt{2}} {6}\\[5pt] &= \underline{\underline{\dfrac{9 \sqrt{2}} {2}}}\end{align}$	f.	$\begin{align}&\dfrac{3} {2\sqrt{5}}\\[5pt] &= \dfrac{3 \times \sqrt{5}} {2\sqrt{5} \times \sqrt{5}}\\[5pt] &= \dfrac{3 \sqrt{5}} {2\times 5}\\[5pt] &= \underline{\underline{\dfrac{3 \sqrt{5}} {10}}}\end{align}$
g.	$\begin{align}&\dfrac{3\sqrt{5}} {2\sqrt{7}}\\[5pt] &= \dfrac{3\sqrt{5} \times \sqrt{7}} {2\sqrt{7} \times \sqrt{7}}\\[5pt] &= \dfrac{3\sqrt{35}} {2 \times 7}\\[5pt] &= \underline{\underline{\dfrac{3 \sqrt{35}} {14}}}\end{align}$	h.	$\begin{align}&\dfrac{2\sqrt3} {3\sqrt{2}}\\[5pt] &= \dfrac{2\sqrt{3} \times \sqrt{2}} {3\sqrt{2} \times \sqrt{2}}\\[5pt] &= \dfrac{2\sqrt{6}} {3\times 2}\\[5pt] &= \underline{\underline{\dfrac{ \sqrt{6}} {3}}}\end{align}$	i.	$\begin{align}&\dfrac{3\sqrt{3}} {2\sqrt{5}}\\[5pt] &= \dfrac{3\sqrt{3} \times \sqrt{5}} {2\sqrt{5} \times \sqrt{5}}\\[5pt] &= \dfrac{3\sqrt{15}} {2\times 5}\\[5pt] &= \underline{\underline{\dfrac{3 \sqrt{15}} {10}}}\end{align}$

a.	$\begin{align}&3 \sqrt{2} \times 2 \sqrt{3}\\[5pt] &= 3 \times 2 \times\sqrt{2} \times \sqrt{3}\\[5pt] &=6\times\sqrt {2\times3}\\[5pt] &=\underline{\underline{6 \sqrt{6}}}\end{align}$	b.	$\begin{align}&5 \sqrt{11} \times 3 \sqrt{7}\\[5pt] &= 5 \times 3 \times\sqrt{11} \times \sqrt{7}\\[5pt] &=15\times\sqrt {11\times7}\\[5pt] &=\underline{\underline{15 \sqrt{77}}}\end{align}$	c.	$\begin{align}&\sqrt{5} \times 3 \sqrt{3}\\[5pt] &= 3 \times\sqrt{5} \times \sqrt{3}\\[5pt] &=\underline{\underline{3 \sqrt{15}}}\end{align}$
d.	$\begin{align}&4 \sqrt{7} \div 2 \sqrt{14}\\[5pt] &= \dfrac{4 \sqrt{7}} {2 \sqrt{14}}\\[5pt] &= \dfrac{2 \sqrt{7}} {\sqrt{ 2\times 7}}\\[5pt] &= \dfrac{2 \sqrt{7}} {\sqrt{ 2}\times \sqrt{7}}\\[5pt] &= \dfrac{2} {\sqrt{ 2}}\\[5pt] &= \dfrac{2 \sqrt{2}} {\sqrt{ 2}\times \sqrt{2}}\\[5pt] &= \dfrac{2 \times \sqrt{2}} {\sqrt{2}}\\[5pt] &=\underline{\underline{\sqrt{2}}}\end{align}$	e.	$\begin{align}&6 \sqrt{27} \div 3 \sqrt{3}\\[5pt] &= \dfrac{6 \sqrt{27}} {3 \sqrt {3}}\\[5pt] &= \dfrac{2 \sqrt{9\times 3}} {\sqrt{3}}\\[5pt] &= \dfrac{2 \times 3 \sqrt{3}} {\sqrt{ 3}}\\[5pt] &=\underline{\underline{6}}\end{align}$	f.	$\begin{align}&\sqrt{48} \div 5 \sqrt{3}\\[5pt] &= \dfrac{\sqrt{16\times 3}} {5\sqrt{ 3}}\\[5pt] &= \dfrac{\sqrt{ 16}\times \sqrt{3}} {5 \sqrt{3}}\\[5pt] &= \dfrac{4 \times \sqrt{3}} {5 \sqrt{3}}\\[5pt] &= \underline{\underline{\dfrac{4} {5}}}\end{align}$

a.	$\begin{align}&2\sqrt{27}-3\sqrt{3}+4\sqrt{7}+3\sqrt{28}\\[5pt] &=2\sqrt{9 \times 3}-3\sqrt{3}+4\sqrt{7}+3\sqrt{4 \times 7}\\[5pt] &=2\times \sqrt{9} \times\sqrt{3}-3\sqrt{3}+4\sqrt{7}+3 \times\sqrt{4}\times\sqrt 7\\[5pt] &=2\times3\times \sqrt{3} -3\sqrt{3}+4\sqrt{7}+3 \times2\times\sqrt {7}\\[5pt] &=6\sqrt{3}-3\sqrt{3}+4\sqrt{7}+6\sqrt 7\\[5pt] &=\underline{\underline{3\sqrt{3}+10\sqrt {7}}}\end{align}$
b.	$\begin{align}&3\sqrt{63}-2\sqrt{7}+3\sqrt{27}+3\sqrt{3}\\[5pt] &=3\sqrt{9 \times 7}-2\sqrt{7}+3\sqrt{9 \times 3}+3\sqrt{3}\\[5pt] &=3\times \sqrt{9} \times\sqrt{7}-2\sqrt{7}+3\times \sqrt{9}\times\sqrt{3}+3\sqrt {3}\\[5pt] &=3\times3\times \sqrt{7} -2\sqrt{7}+3\times3\times\sqrt{3}+3\sqrt {3}\\[5pt] &=9\sqrt{7}-2\sqrt{7}+9\sqrt{3}+3\sqrt {3}\\[5pt] &=\underline{\underline{7\sqrt{7}+12\sqrt{3}}}\end{align}$
c.	$\begin{align}&2\sqrt{128}-3\sqrt{50}+2\sqrt{162}+\dfrac{4} {\sqrt{ 2}}\\[5pt] &=2\sqrt{64 \times 2}-3\sqrt{25 \times 2}+2 \sqrt{81\times2}+\dfrac{4 \times \sqrt{2}} {\sqrt{ 2} \times \sqrt{2}}\\[5pt] &=2\times \sqrt{64}\times \sqrt{2}-3\times\sqrt{25}\times\sqrt{2}+2\times\sqrt{81}\times\sqrt {2}+ \dfrac{4\times\sqrt{2}}{2}\\[5pt] &=2\times8\times \sqrt{2} -3\times5\times\sqrt{2}+2\times9\times\sqrt{2}+2\sqrt {2}\\[5pt] &=16\sqrt{2}-15\sqrt{2}+18\sqrt{2}+2\sqrt{2}\\[5pt] &=\underline{\underline{21\sqrt{2}}}\\[20pt]\end{align}$
d.	$\begin{align}&\sqrt{99}-2\sqrt{44}+\dfrac{110} {\sqrt{44}}\\[5pt] &=\sqrt{9 \times 11}-2\sqrt{4 \times 11}+\dfrac{110} {\sqrt{ 4 \times 11}}\\[5pt] &=\sqrt{9}\times \sqrt{11}-2\times\sqrt{4}\times\sqrt{11}+\dfrac{110}{2\sqrt{11}}\\[5pt] &=3\times \sqrt{11} -2\times2\times\sqrt{11}+\dfrac{55 \times \sqrt{11}} {\sqrt{11} \times \sqrt{11}}\\[5pt] &=3\sqrt{11}-4\sqrt{11}+\dfrac{55 \times \sqrt{11}} {11}\\[5pt] &=3\sqrt{11}-4\sqrt{11}+5 \sqrt{11}\\[5pt] &=\underline{\underline{4\sqrt{11}}}\end{align}$
e.	$\begin{align}&\dfrac{\sqrt{20}}{2} - \sqrt{5}\\[5pt] &=\dfrac{\sqrt{4 \times 5}} {2} - \sqrt{5}\\[5pt] &=\dfrac{2\sqrt{5}}{2} - \sqrt{5}\\[5pt] &=\sqrt{5} - \sqrt{5}\\[5pt] &=\underline{\underline{0}}\\[20pt]\end{align}$

By studying this lesson will be able

to investigate number sets, workout basic mathematical regarding surds.