Nine Standard >> Operations on surds | Part-4

1. Simplify \(4\sqrt{3}-3\sqrt{12}+2\sqrt{75}\)
=\(4\sqrt{3}-3\sqrt{2 \times 2 \times3}+2\sqrt{5 \times 5 \times 3}\)

Solution:
  \(4\sqrt{3}-3\sqrt{12}+2\sqrt{75}\)
=\(4\sqrt{3}-3\sqrt{2 \times 2 \times3}+2\sqrt{5 \times 5 \times 3}\)
=\(4\sqrt{3}-3\sqrt{2^{2} \times3}+2\sqrt{5^{2} \times 3}\)
=\(4\sqrt{3}-3 \times 2\sqrt{3}+2 \times 5 \sqrt{3}\)
=\((4-6+10)\sqrt{3}\)
=\((14-6)\sqrt{3}\)
=\(8\sqrt{3}\)

2. Simplify \(\sqrt[4]{81}\)-8\(\sqrt[3]{216}\)+15\(\sqrt[5]{32}\)+\(\sqrt{225}\)

Solution:
\(\sqrt[4]{81}\)=\(\sqrt[4]{3 \times 3 \times 3 \times 3}\)=\(\sqrt[4]{3^{4}}\)=3
\(\sqrt[3]{216}\)=\(\sqrt[3]{6 \times 6 \times 6}\)=\(\sqrt[3]{6^{3}}\)=6  
\(\sqrt[5]{32}\)=\(\sqrt[5]{2 \times 2 \times 2 \times 2 \times 2}\)=\(\sqrt[5]{2^{5}}\)=2  
\(\sqrt{225}\)=\(\sqrt{5 \times 3 \times 5 \times 3 }\)=\(\sqrt{15^{2}}\)=15

\(\sqrt[4]{81}\)-8\(\sqrt[3]{216}\)+15\(\sqrt[5]{32}\)+\(\sqrt{225}\)
=\(3-8 \times 6 +15 \times 2+15\)
=3-48+30+15
=48-48=0

3. Simplify \(5\sqrt{147}-\frac{4}{3}\sqrt{\frac{1}{3}}+7\sqrt{\frac{1}{3}}\)

Solution:
\(5\sqrt{147}\)=\(5\sqrt{7^{2} \times 3}\)
       =\((5 \times 7)\sqrt{3}=35\sqrt{3}\)
\(-\frac{4}{3}\sqrt{\frac{1}{3}}=-\frac{4}{3}\sqrt{\frac{1 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}}}\)
  =\(-\frac{4\sqrt{3}}{9}\)
\(7\sqrt{\frac{1}{3}}=7 \times \frac{1 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}}\)
  =\(\frac{7\sqrt{3}}{3}\)

 \(5\sqrt{147}-\frac{4}{3}\sqrt{\frac{1}{3}}+7\sqrt{\frac{1}{3}}\)

=\(35\sqrt{3}-\frac{4\sqrt{3}}{9}+7\sqrt{\frac{1}{3}}\)
=\((35-\frac{4}{9}+\frac{7}{3})\sqrt{3}\)
=\(\Big(\frac{315-4+21}{9}\Big)\sqrt{3}\)
=\(\Big(\frac{336-4}{9}\Big)\sqrt{3}\)
=\(\frac{332}{9}\sqrt{3}\)