(300000)³ x (2000)⁴ =(3.0×10⁵)³×(2.0×10³) ⁴
=(3.0)³ ×(10⁵)³×(2.0)⁴ ×(10³)⁴
=(27.0)×(10¹⁵)×(16.0)×(10¹²)
=(2.7×10¹)×(10¹⁵)×(1.6 × 10¹)×(10¹² )
=2.7x1.6 x10¹x 10¹⁵ x10¹x10¹²
=4.32 x 10 ^1+15+1+12=4.32 x 10²⁹

(i) விகிதப்படுத்தும் காரணியான \( \sqrt{14}\) ஆல் பகுதி மற்றும் தொகுதியைப் பெருக்குக.
\({7\over \sqrt{14}}={7\over\sqrt{14}}\times {7\sqrt{14}\over \sqrt{14}}={\sqrt{14}\over 2}\)
(ii) \({5+\sqrt{3}\over 5-\sqrt{3}}={(5+\sqrt{3})\over(5-\sqrt{3})}\times {(5+\sqrt{3})\over(5+\sqrt{3})}={(5+\sqrt{3})^2\over 5^2-(\sqrt{3})^2}\)
\(={5^2+(\sqrt{3})^2+2\times 5\times \sqrt{3}\over 25-3}\)
\(={25+3+10\sqrt{3}\over 22}={28+10\sqrt{3}\over22}={2\times [14+5\sqrt{3}]\over 22}\)
\(={14+5\sqrt{3}\over 11}\)

\(\sqrt[3]{40} \times \sqrt[3]{16}=(\sqrt[3]{2\times2\times2\times5})\times(\sqrt[3]{2\times2\times2\times2})\)
\(=(2\times \sqrt[3]{5})\times (2\times \sqrt[3]{2})=4\times (\sqrt[3]{2}\times \sqrt[3]{5})=4\times \sqrt[3]{2\times 5}\)
\(=4\sqrt[3]{10}\)

\(\sqrt[3]{2},\sqrt[2]{4},\sqrt[4]{3}\) ஆகியவற்றின் வரிசைகள் 3,2.4
3,2,4 இன் மீ.பொ.ம =12.
\(\sqrt[3]{2}=(2^{1\over3})=({2^{4\over12}})=\sqrt[12]{2^4}=\sqrt[12]{16};\)
\(\sqrt[2]{4}=(4^{1\over2})=({4^{6\over12}})=\sqrt[12]{4^6} =\sqrt[12]{4096}\)
\(\sqrt[4]{3}=(3^{1\over4})=(3^{3\over12})=\sqrt[12]{3^3}=\sqrt[12]{27}\)
\(\sqrt[3]{2},\sqrt[4]{3},\sqrt[2]{4}\) ஆனது \(\sqrt[12]{16}<\sqrt[12]{27}<\sqrt[12]{4096}\)
ஏறுவரிசை \(\sqrt[3]{2},\sqrt[4]{3},\sqrt[2]{4}\)

\(\sqrt[3]{192}=\sqrt[3]{4\times 4\times 4\times 3}=4\sqrt[3]{3}\)

(i)\(\begin{aligned} \sqrt{5} & =5^{\frac{1}{2}} \end{aligned}\)
(ii) \(\begin{aligned} \sqrt[2]{7} & =7^{\frac{1}{2}} \end{aligned}\)
(iii) \(\begin{aligned} (\sqrt[3]{49})^5 & =49^{\frac{5}{3}}=\left(7^2\right)^{\frac{5}{3}}=7^{\frac{10}{3}} \\ \end{aligned}\)
(iv)\(\begin{aligned} \left(\frac{1}{\sqrt[3]{100}}\right)^7 & =\left(\frac{1}{100^{\frac{1}{3}}}\right)^7=\left(\frac{1^{\frac{1}{3}}}{100^{\frac{1}{3}}}\right)^7=\left(\frac{1}{100}\right)^{\frac{7}{3}}=\left(\frac{1}{10^2}\right)^{\frac{7}{3}}=\left(10^{-2}\right)^{\frac{7}{3}}=10^{\frac{-14}{3}} \end{aligned}\)

(i) \((49)^{\frac{1}{2}}=\sqrt{49}=\sqrt{7 \times 7}^{1/2}=7 \)
(ii) \((243)^{\frac{2}{5}}=\left(3^5\right)^{\frac{2}{5}}=3^{\frac{\not 5 \times 2}{\not 8}}=3^2=9 \)
(iii) \(9^{\frac{-3}{2}}=\left(3^2\right)^{\frac{-3}{2}}=\left(3\right)^{\not 2 \times \frac{-3}{\not 2}}=3^{-3}=\frac{1}{3^3}=\frac{1}{27} \)
(iv) \(\left(\frac{64}{125}\right)^{\frac{-2}{3}}=\left(\frac{125}{64}\right)^{\frac{2}{3}}=\left(\left(\frac{5}{4}\right)^3\right)^{\frac{2}{3}}=\left(\frac{5}{4}\right)^{\not 3 \times \frac{2}{\not 3}}=\left(\frac{5}{4}\right)^2=\frac{25}{16}\)

(i) 625 = 5⁴

(ii) \(\frac{1}{5}\)
\(\frac{1}{5}\)=5^-1
(iii) \(\sqrt{5}\)=5^{\(\frac{1}{2}\)}
(iv) \(\sqrt{125}=\sqrt{5^3}=\left(5^3\right)^{\frac{1}{2}}=5^{\frac{3}{2}}\)

\(\sqrt{8}=\sqrt{2}\times\sqrt2 \times \sqrt2=({2}^{1\over2})^3\). இதனை \(2^{3\over2}\)எனவும் எழுதலாம்.

32 = 2 x 2 x 2 x 2 x 2 = 2⁵

\(({1\over3})^{-2}={1\over ({1\over3})^2}={1\over ({1\over9})}=9\)

(i) \(\begin{aligned} (300000)^2 \times(20000)^4 & =\left(3.0 \times 10^5\right)^2 \times\left(2.0 \times 10^4\right)^4=3^2 \times 10^{10} \times 2^4 \times 10^{16} \\ \end{aligned}\)
\(\begin{aligned} =9 \times 16 \times 10^{10+16}=144 \times 10^{26}=1.44 \times 10^{28} \end{aligned}\)
(ii) (0.000001)¹¹\(\div\)(0.005)³
\(\begin{aligned} \frac{(0.000001)^{11}}{(0.005)^3} & =\frac{(0.000001)^{11}}{(0.005)^3}=\frac{\left(1.0 \times 10^{-6}\right)^{11}}{\left(5.0 \times 10^{-3}\right)^3}=\frac{1.0 \times 10^{-66}}{125.0 \times 10^{-9}}=\frac{1.0 \times 10^{-66}}{1.25 \times 10^2 \times 10^{-9}} \\ \end{aligned}\)
\(\begin{aligned} =\frac{1.0}{1.25} \times 10^{-66-2+9}=0.8 \times 10^{-59}=8.0 \times 10^{-1} \times 10^{-59}=8.0 \times 10^{-60} \end{aligned}\)
(iii) \(\begin{aligned} \left\{(0.00003)^6 \times(0.00005)^4\right\} & \div\left\{(0.009)^3 \times(0.05)^2\right\} \end{aligned}\)
\(\begin{aligned} =\frac{\left(3.0 \times 10^{-5}\right)^6 \times\left(5.0 \times 10^{-5}\right)^4}{\left(9.0 \times 10^{-3}\right)^3 \times\left(5.0 \times 10^{-2}\right)^2}=\frac{3^6 \times 10^{-30} \times 5^4 \times 10^{-20}}{9^3 \times 10^{-9} \times 5^2 \times 10^{-4}} \\ \end{aligned}\)
\(\begin{aligned} =\frac{3^6 \times 5^4 \times 10^{-30-20}}{\left(3^2\right)^3 \times 10^{-9-4} \times 5^2}=\frac{\not 3^ {\not 6} \times 5^4 \times 10^{-50}}{\not 3^ {\not 6} \times 5^2 \times 10^{-13}} \\ \end{aligned}\)
= 5^4-2 \(\times\) 10^-50+13 = 5² \(\times\)10^-37 = 25 \(\times\) 10^-37
= 2.5 \(\times\) 10¹ \(\times\) 10^-37 = 2.5 \(\times\) 10^-36

\({2\sqrt{6}-\sqrt{5}\over 3\sqrt{5}-2\sqrt{6}}\)
\(=\frac{2 \sqrt{6}-\sqrt{5}}{3 \sqrt{5}-2 \sqrt{6}} \times \frac{3 \sqrt{5}+2 \sqrt{6}}{3 \sqrt{5}-2 \sqrt{6}} \)
\(=\frac{6 \sqrt{30}+4 \times 6-3 \times 5-2 \sqrt{30}}{9 \times 5-4 \times 6} \)
\(=\frac{4 \sqrt{30}+24-15}{45-24}=\frac{9+4 \sqrt{30}}{21} \)

(i) \(\begin{aligned} \sqrt{40}-\sqrt{20} & =\sqrt{2^2 \times 10}-\sqrt{2^2 \times 5} \end{aligned}\)
\(\begin{aligned} =(2 \times \sqrt{10})-(2 \times \sqrt{5}) \end{aligned}\)
\(\begin{aligned} =(2 \times 3.162)-(2 \times 2.236) \end{aligned}\)
= 6.324 - 4.472
= 1.852
(ii) \(\begin{aligned} \sqrt{300}+\sqrt{90}-\sqrt{8} & =\sqrt{3 \times 100}+\sqrt{9 \times 10}+\sqrt{4 \times 2}=10 \sqrt{3}+3 \sqrt{10}-2 \sqrt{2} \\ \end{aligned}\)
\( \begin{aligned} =10 \times 1.732+3 \times 3.162-2 \times 1.414 \end{aligned}\)
= 17.32 + 9.486 - 2.828 = 23.978

(i) \(\sqrt{3}=3^{1\over2}\)
\(=3^{6\over12}\)
\(=\sqrt[12]{3^6}\)
\(=\sqrt[12]{729}\)
(ii) \(\sqrt[4]{3}=3^{1\over4}\)
\(=3^{3\over12}\)
\(=\sqrt[12]{3^3}\)
\(=\sqrt[12]{27}\)
(iii) \(\sqrt[3]{3}=3^{1\over3}\)
\(=3^{4\over12}\)
\(=\sqrt[12]{3^4}\)
\(=\sqrt[12]{81}\)

(i) \(\sqrt[3]{5}, \sqrt[9]{4}, \sqrt[6]{3}\)
\(\sqrt[3]{5}, \sqrt[9]{4}, \sqrt[6]{3}\) ஆகியவற்றின் வரிசைகள் 3, 9, 6. 3, 9, 6 ன் மீ.பொ.ம = 18
\(\begin{aligned} \sqrt[3]{5} & =\sqrt[3 \times 6]{5^6}=\sqrt[18]{15625} \end{aligned}\)
\(\begin{aligned} \sqrt[9]{4} & =\sqrt[9 \times 2]{4^2}=\sqrt[18]{16} \end{aligned}\)
\(\begin{aligned} \sqrt[6]{3} & =\sqrt[6 \times 3]{3^3}=\sqrt[18]{27} \end{aligned}\)
\(\begin{aligned} \sqrt[18]{15625} & >\sqrt[18]{27}>\sqrt[18]{16} \end{aligned}\)
\(\begin{aligned} \text { i.e, } \sqrt[3]{5} & >\sqrt[6]{3}>\sqrt[9]{4} \end{aligned}\)

(ii) \(\sqrt[2]{\sqrt[3]{5}}, \sqrt[3]{\sqrt[4]{7}}, \sqrt{\sqrt{3}}\)
\(\sqrt[2]{\sqrt[3]{5}}, \sqrt[3]{\sqrt[4]{7}}, \sqrt{\sqrt{3}}\) ஆகியவற்றின் வரிசைகள் 6, 12, 4
6, 12, 4 ன் மீ.பொ.ம = 12
\(\begin{aligned} \sqrt[2]{\sqrt[3]{5}} & =\sqrt[6]{5}=\sqrt[6 \times 2]{5^2}=\sqrt[12]{25} \\ \end{aligned}\)
\(\begin{aligned} \sqrt[3]{\sqrt[4]{7}} & =\sqrt[12]{7}=\sqrt[12]{7^1}=\sqrt[12]{7} \end{aligned}\)
\(\begin{aligned} \sqrt{\sqrt{3}} & =\sqrt[4]{3}=\sqrt[4 \times 3]{3^3}=\sqrt[12]{27} \end{aligned}\)
\(\begin{aligned} \sqrt[12]{27} & >\sqrt[12]{25}>\sqrt[12]{7} \end{aligned}\)
\(\begin{aligned} \text { i.e, } \sqrt{\sqrt{3}} & >\sqrt[2]{\sqrt[3]{5}}>\sqrt[3]{\sqrt[4]{7}} \end{aligned}\)

(i) \(5 \sqrt{3}+18 \sqrt{3}-2 \sqrt{3}=(5+18-2) \sqrt{3}=21 \sqrt{3}\)
(ii) \(4 \sqrt[3]{5}+2 \sqrt[3]{5}-3 \sqrt[3]{5}=(4+2-3) \sqrt[3]{5}=3 \sqrt[3]{5}\)
(iii) \(3 \sqrt{75}+5 \sqrt{48}-\sqrt{243}\)
\(=3 \sqrt{5 \times 5 \times 3}+5 \sqrt{3 \times 2 \times 2 \times 2 \times 2}-\sqrt{3 \times 3 \times 3 \times 3 \times 3}\)
\(\begin{aligned} & =\quad 3 \times 5 \sqrt{3}+5 \times 2 \times 2 \sqrt{3}-3 \times 3 \sqrt{3} \\ \end{aligned}\)
\(\begin{aligned} & =\quad 15 \sqrt{3}+20 \sqrt{3}-9 \sqrt{3} \end{aligned}\)
\(\begin{aligned} & =\quad(15+20-9) \sqrt{3} \end{aligned}\)
\(\begin{aligned} =\quad 26 \sqrt{3} \end{aligned}\)

(iv) \(\begin{array}{ll} 5 \sqrt[3]{40}+2 \sqrt[3]{625}-3 \sqrt[3]{320} \end{array}\)
\(\begin{array}{ll} = & 5 \sqrt[3]{2^3 \times 5}+2 \sqrt[3]{5^3 \times 5}-3 \sqrt[3]{2^3 \times 2^3 \times 5} \end{array}\)
\(\begin{array}{ll} = & 5 \times 2 \times \sqrt[3]{5}+2 \times 5 \sqrt[3]{5}-3 \times 2 \times 2 \sqrt[3]{5} \\ \end{array}\)
\(\begin{array}{ll} = & 10 \sqrt[3]{5}+10 \sqrt[3]{5}-12 \sqrt[3]{5} \end{array}\)
\(\begin{array}{ll} = & (10+10-12) \sqrt[3]{5} \end{array}\)
\(\begin{array}{ll} =\quad & 8 \sqrt[3]{5} \end{array}\)

l = 5 x 10⁵
b = 4 x 10⁴ 
பரப்பளவு = l x b 
5 x 10⁵ x 4 x 10⁴ 
= 20 x 10⁵⁺⁴
= 20 x 10⁹ = 2.0 x 10¹ x 10⁹ = 2 x 10¹⁰மீ²

\((2 \sqrt{5}-\sqrt{2})^2=4 \times 5-2 \times 2 \sqrt{5} \times \sqrt{2}+2 \)
\(20-4 \sqrt{10}+2=22-4 \sqrt{10} \)

\(\left(\because \sqrt{80}=\sqrt{16 \times 5}=\sqrt{4^2 \times 5}=4 \sqrt{5}\right)\)

\(\sqrt{25}=\pm5\) என எழுதுவது தவறு.