Surds And Indices

$${\left( {\sqrt 8 } \right)^{\frac{1}{3}}}$$
$$ = {\left( {{8^{\frac{1}{2}}}} \right)^{\frac{1}{3}}}$$
$$ = {8^{\left( {\frac{1}{2} \times \frac{1}{3}} \right)}}$$
$$ = {8^{\frac{1}{6}}}$$
$$ = {\left( {{2^3}} \right)^{\frac{1}{6}}}$$
$$ = {2^{\left( {3 \times \frac{1}{6}} \right)}}$$
$$ = {2^{\frac{1}{2}}}$$
$$ = \sqrt 2 $$

$${\left( {\frac{{32}}{{243}}} \right)^{ - \frac{4}{5}}}$$
$${\text{ = }}{\left\{ {{{\left( {\frac{2}{3}} \right)}^5}} \right\}^{ - \frac{4}{5}}}$$
$${\text{ = }}{\left( {\frac{2}{3}} \right)^{5 \times \frac{{\left( { - 4} \right)}}{5}}}$$
$${\text{ = }}{\left( {\frac{2}{3}} \right)^{\left( { - 4} \right)}}$$
$${\text{ = }}{\left( {\frac{3}{2}} \right)^4}$$
$${\text{ = }}\frac{{{3^4}}}{{{2^4}}}$$
$${\text{ = }}\frac{{81}}{{16}}{\text{ }}$$

$${\text{ 2}}{{\text{7}}^{ - \frac{2}{3}}}$$
$$ = {\left( {{3^3}} \right)^{ - \frac{2}{3}}}$$
$$ = {3^{\left[ {3 \times \left( { - \frac{2}{3}} \right)} \right]}}$$
$$ = {3^{ - 2}}$$
$$ = \frac{1}{{{3^2}}}$$
$$ = \frac{1}{9} $$
$$ {\text{Clearly}},\,\,0 < \frac{1}{9} < 1 \cr} $$

$$\frac{1}{{\sqrt {3.25} + \sqrt {2.25} }} \times \frac{{\sqrt {3.25} - \sqrt {2.25} }}{{\sqrt {3.25} - \sqrt {2.25} }}$$
$$ = \frac{{\sqrt {3.25} - \sqrt {2.25} }}{{3.25 - 2.25}}$$
$$ = \sqrt {3.25} - \sqrt {2.25} \,......(i)$$
$${\text{Similarly}}$$
$$\frac{1}{{\sqrt {4.25} + \sqrt {3.25} }}$$
$$ = \sqrt {4.25} - \sqrt {3.25} \,.......(ii)$$
$$\frac{1}{{\sqrt {5.25} + \sqrt {4.25} }}$$
$$ = \sqrt {5.25} - \sqrt {4.25} \,.......(iii)$$
$$\frac{1}{{\sqrt {6.25} + \sqrt {5.25} }}$$
$$ = \sqrt {6.25} - \sqrt {5.25} \,.......(iv)$$
$${\text{Now}}\,{\text{add}}\,{\text{all}}\,{\text{them}}$$
$$\sqrt {3.25} - \sqrt {2.25} + \sqrt {4.25} - \sqrt {3.25} + \sqrt {5.25} - \sqrt {4.25} + \sqrt {6.25} - \sqrt {5.25} $$
$$ = \sqrt {6.25} - \sqrt {2.25} $$
$$ = 2.5 - 1.5$$
$$ = 1$$

$$\frac{{{3^0} + {3^{ - 1}}}}{{{3^{ - 1}} - {3^0}}}$$
$$ = \frac{{1 + \frac{1}{3}}}{{\frac{1}{3} - 1}}$$
$$ = \frac{{\frac{4}{3}}}{{ - \frac{2}{3}}}$$
$$ = - 2$$

$${\text{Let }}\,{6^{1.2}} \times {36^x} \times {30^{2.4}} \times {25^{1.3}} = {30^5}$$
$${\text{Then,}}\,{6^{1.2}} \times {({6^2})^x} \times {(6 \times 5)^{2.4}} \times {({5^2})^{1.3}} = {30^5}$$
$$ ⇒ {6^{1.2}} \times {6^{2x}} \times {6^{2.4}} \times {5^{2.4}} \times {5^{2.6}} = {(6 \times 5)^5}$$
$$ ⇒ {6^{\left( {1.2 + 2x + 2.4} \right)}} \times {5^{\left( {2.4 + 2.6} \right)}} = {6^5} \times {5^5}$$
$$ ⇒ {6^{\left( {3.6 + 2x} \right)}} \times {5^5} = {6^5} \times {5^5}$$
$$ ⇒ 3.6 + 2x = 5$$
$$ ⇒ 2x = 1.4$$
$$ ⇒ x = 0.7$$

$$\left[ {\frac{{\sqrt 3 + \sqrt 2 }}{{\sqrt 3 - \sqrt 2 }} - \frac{{\sqrt 3 - \sqrt 2 }}{{\sqrt 3 + \sqrt 2 }}} \right]$$
$$ = \frac{{\sqrt 3 + \sqrt 2 }}{{\sqrt 3 - \sqrt 2 }} \times $$   $$\frac{{\sqrt 3 + \sqrt 2 }}{{\sqrt 3 + \sqrt 2 }} - $$   $$\frac{{\sqrt 3 - \sqrt 2 }}{{\sqrt 3 + \sqrt 2 }} \times $$  $$\frac{{\sqrt 3 - \sqrt 2 }}{{\sqrt 3 - \sqrt 2 }}$$
$$ = \frac{{{{\left( {\sqrt 3 + \sqrt 2 } \right)}^2}}}{{3 - 2}} - \frac{{{{\left( {\sqrt 3 - \sqrt 2 } \right)}^2}}}{{3 - 2}}$$
$$ = \left( {3 + 2 + 2\sqrt 6 } \right) - \left( {3 + 2 - 2\sqrt 6 } \right)$$
$$ = 4\sqrt 6 {\text{ }}$$

$$\sqrt 3 = 1.732$$
$$ ⇒ \frac{{2 + \sqrt 3 }}{{2 - \sqrt 3 }} \times \frac{{2 + \sqrt 3 }}{{2 + \sqrt 3 }}$$
$$ ⇒ \frac{{{{\left( {2 + \sqrt 3 } \right)}^2}}}{{4 - 3}}$$
$$ ⇒ 4 + 3 + 4\sqrt 3 $$
$$ ⇒ 7 + 4 \times 1.732$$
$$ ⇒ 7 + 6.928$$
$$ ⇒ 13.928$$

$$16\sqrt {\frac{3}{4}} - 9\sqrt {\frac{4}{3}} $$
$$ ⇒ 16\sqrt {\frac{{3 \times 4}}{{4 \times 4}}} - 9\sqrt {\frac{{3 \times 4}}{{3 \times 3}}} $$
$$ ⇒ 16 \times \frac{{\sqrt {12} }}{4} - \frac{{9\sqrt {12} }}{3}$$
$$ ⇒ 4\sqrt {12} - 3\sqrt {12} $$
$$ ⇒ \sqrt {12} $$
$$ ⇒ 3.46$$

$${\text{Let }}{2^{3.6}} \times {4^{3.6}} \times {4^{3.6}} \times {(32)^{2.3}} = {\left( {32} \right)^x}$$
$${\text{Then,}}{2^{3.6}} \times {\left( {{2^2}} \right)^{3.6}} \times {\left( {{2^2}} \right)^{3.6}} \times {({2^5})^{2.3}} = {\left( {{2^5}} \right)^x}$$
$$ ⇒ {2^{3.6}} \times {2^{\left( {2 \times 3.6} \right)}} \times {2^{\left( {2 \times 3.6} \right)}} \times {({2^5})^{2.3}} = {\left( {{2^5}} \right)^x}$$
$$ ⇒ {2^{\left( {3.6 + 7.2 + 7.2} \right)}} \times {({2^5})^{2.3}} = {\left( {{2^5}} \right)^x}$$
$$ ⇒ {2^{18}} \times {({2^5})^{2.3}} = {\left( {{2^5}} \right)^x}$$
$$ ⇒ {\left( {{2^5}} \right)^{3.6}} \times {({2^5})^{2.3}} = {\left( {{2^5}} \right)^x}$$
$$ ⇒ {\left( {{2^5}} \right)^{\left( {3.6 + 2.3} \right)}} = {\left( {{2^5}} \right)^x}$$
$$ ⇒ {\left( {{2^5}} \right)^{5.9}} = {\left( {{2^5}} \right)^x}$$
$$ ⇒ x = 5.9$$