Surds And Indices

$$\frac{{\frac{3}{{2 + \sqrt 3 }} - \frac{2}{{2 - \sqrt 3 }}}}{{2 - 5\sqrt 3 }}$$
$$ = \frac{{\frac{{3\left( {2 - \sqrt 3 } \right) - 2\left( {2 + \sqrt 3 } \right)}}{{\left( {2 + \sqrt 3 \,} \right)\left( {2 - \sqrt 3 } \right)}}}}{{2 - 5\sqrt 3 }}$$
$$ = \frac{{6 - 3\sqrt 3 - 4 - 2\sqrt 3 }}{{\left( {2 + \sqrt 3 } \right)\left( {2 - \sqrt 3 } \right)\left( {2 - 5\sqrt 3 } \right)}}$$
$$ = \frac{{2 - 5\sqrt 3 }}{{2 - 5\sqrt 3 }}$$
$$ = 1$$

$$\left( {\sqrt 8 - \sqrt 4 - \sqrt 2 } \right)$$
$$ = 2\sqrt 2 - 2 - \sqrt 2 $$
$$ = 2\sqrt 2 - \sqrt 2 - 2$$
$$ = \sqrt 2 - 2$$

$${\text{6}}{{\text{4}}^{ - \frac{2}{3}}} \times {\left( {\frac{1}{4}} \right)^{ - 2}}$$
$$ = {\left( {{4^3}} \right)^{ - \frac{2}{3}}} \times {\left( {\frac{1}{4}} \right)^{ - 2}}$$
$$ = {4^{ - 2}} \times {\left( {\frac{1}{4}} \right)^{ - 2}}$$
$$ = {\left( {\frac{1}{4}} \right)^2} \times {\left( {\frac{1}{4}} \right)^{ - 2}}$$
$$ = {\left( {\frac{1}{4}} \right)^{2 - 2}}$$
$$ = {\left( {\frac{1}{4}} \right)^0}$$
$$ = 1$$

$$\left( {\frac{{{9^2} \times {{18}^4}}}{{{3^{16}}}}} \right)$$
$$ = \frac{{{9^2} \times {{\left( {9 \times 2} \right)}^4}}}{{{3^{16}}}}$$
$$ = \frac{{{{\left( {{3^2}} \right)}^2} \times {{\left( {{3^2}} \right)}^4} \times {2^4}}}{{{3^{16}}}}$$
$$ = \frac{{{3^4} \times {3^8} \times {2^4}}}{{{3^{16}}}}$$
$$ = \frac{{{3^{\left( {4 + 8} \right)}} \times {2^4}}}{{{3^{16}}}}$$
$$ = \frac{{{3^{12}} \times {2^4}}}{{{3^{16}}}}$$
$$ = \frac{{{2^4}}}{{{3^{\left( {16 - 12} \right)}}}}$$
$$ = \frac{{{2^4}}}{{{3^4}}}$$
$$ = \frac{{16}}{{81}}$$

$$\frac{{{4^3} \times {5^4}}}{{{4^5}}}$$
$$ = \frac{{{5^4}}}{{{4^{\left( {5 - 3} \right)}}}}$$
$$ = \frac{{{5^4}}}{{{4^2}}}$$
$$ = \frac{{625}}{{16}}$$
$$ = 39.0625$$

$$\frac{2}{{\sqrt 7 + \sqrt 5 }} + \frac{7}{{\sqrt {12} - \sqrt 5 }} - \frac{5}{{\sqrt {12} - \sqrt 7 }}$$
$$ = \frac{2}{{\sqrt 7 + \sqrt 5 }} \times $$  $$\frac{{\sqrt 7 - \sqrt 5 }}{{\sqrt 7 - \sqrt 5 }} + $$  $$\frac{7}{{\sqrt {12} - \sqrt 5 }} \times $$  $$\frac{{\sqrt {12} + \sqrt 5 }}{{\sqrt {12} + \sqrt 5 }} - $$  $$\left( {\frac{5}{{\sqrt {12} - \sqrt 7 }} \times \frac{{\sqrt {12} + \sqrt 7 }}{{\sqrt {12} + \sqrt 7 }}} \right)$$
$$ = \frac{{2\left( {\sqrt 7 - \sqrt 5 } \right)}}{2} + $$   $$\frac{{7\left( {\sqrt {12} + \sqrt 5 } \right)}}{7} - $$   $$\frac{{5\left( {\sqrt {12} + \sqrt 7 } \right)}}{5}$$
$$\eqalign{ &= \sqrt 7 - \sqrt 5 + \sqrt {12} + \sqrt 5 - \sqrt {12} - \sqrt 7 \cr &= 0$$

$$\root 3 \of {{2^4}\sqrt {{2^{ - 5}}\sqrt {{2^6}} } } {\text{ }}$$
$$ = \root 3 \of {{2^4}\sqrt {{2^{ - 5}}{{\left( {{2^6}} \right)}^{\frac{1}{2}}}} } {\text{ }}$$
$$ = \root 3 \of {{2^4}\sqrt {{2^{ - 5}}{{\left( 2 \right)}^{\left( {6 \times \frac{1}{2}} \right)}}} } {\text{ }}$$
$$ = \root 3 \of {{2^4}\sqrt {{2^{ - 5}}{{.2}^3}} } {\text{ }}$$
$$ = \root 3 \of {{2^4}\sqrt {{2^{\left( { - 5 + 3} \right)}}} } {\text{ }}$$
$$ = \root 3 \of {{2^4}\sqrt {{2^{ - 2}}} } {\text{ }}$$
$$ = \root 3 \of {{2^4}.{{\left( {{2^2}} \right)}^{\frac{1}{2}}}} $$
$${\text{ = }}\root 3 \of {{2^4}{{.2}^{\left( { - 2 \times \frac{1}{2}} \right)}}} $$
$$ = \root 3 \of {{2^4}{{.2}^{\left( { - 1} \right)}}} $$
$$ = \root 3 \of {{2^{\left( {4 - 1} \right)}}} $$
$${\text{ = }}\root 3 \of {{2^3}} $$
$${\text{ = }}{\left( {{2^3}} \right)^{\frac{1}{3}}}$$
$${\text{ = }}{{\text{2}}^{\left( {3 \times \frac{1}{3}} \right)}}$$
$${\text{ = 2 }}$$

$$\sqrt {2\sqrt {2\sqrt {2\sqrt {2\sqrt 2 } } } } $$
$$ = \sqrt {2\sqrt {2\sqrt {2\sqrt {{{2.2}^{\frac{1}{2}}}} } } } $$
$$ = \sqrt {2\sqrt {2\sqrt {2\sqrt {{2^{\left( {1 + \frac{1}{2}} \right)}}} } } } $$
$$ = \sqrt {2\sqrt {2\sqrt {2\sqrt {{2^{\frac{3}{2}}}} } } } $$
$$ = \sqrt {2\sqrt {2\sqrt {2.{{\left( {{2^{\frac{3}{2}}}} \right)}^{\frac{1}{2}}}} } } $$
$$ = \sqrt {2\sqrt {2\sqrt {{{2.2}^{\left( {\frac{3}{2} \times \frac{1}{2}} \right)}}} } } $$
$$ = \sqrt {2\sqrt {2\sqrt {{{2.2}^{\frac{3}{4}}}} } } $$
$$ = \sqrt {2\sqrt {2\sqrt {{2^{\left( {1 + \frac{3}{4}} \right)}}} } } $$
$$ = \sqrt {2\sqrt {2\sqrt {{2^{\frac{7}{4}}}} } } $$
$$ = \sqrt {2\sqrt {2{{\left( {{2^{\frac{7}{4}}}} \right)}^{\frac{1}{2}}}} } $$
$$ = \sqrt {2\sqrt {{{2.2}^{\left( {\frac{7}{4} \times \frac{1}{2}} \right)}}} } $$
$$ = \sqrt {2\sqrt {{{2.2}^{\frac{7}{8}}}} } $$
$$ = \sqrt {2\sqrt {{2^{\left( {1 + \frac{7}{8}} \right)}}} } $$
$$ = \sqrt {2\sqrt {{2^{\frac{{15}}{8}}}} } $$
$$ = \sqrt {2{{\left( {{2^{\frac{{15}}{8}}}} \right)}^{\frac{1}{2}}}} $$
$$ = \sqrt {{{2.2}^{\frac{{15}}{{16}}}}} $$
$$ = \sqrt {{2^{\left( {1 + \frac{{15}}{{16}}} \right)}}} $$
$$ = \sqrt {{2^{\frac{{31}}{{16}}}}} $$
$$ = {\left( {{2^{\frac{{31}}{{16}}}}} \right)^{\frac{1}{2}}}$$
$$ = {2^{\left( {\frac{{31}}{{16}} \times \frac{1}{2}} \right)}}$$
$$ = {2^{\frac{{31}}{{32}}}}$$

$${\left( {0.03125} \right)^{ - \frac{2}{5}}}$$
$$ = {\left[ {{{\left( {0.5} \right)}^5}} \right]^{ - \frac{2}{5}}}$$
$$ = {0.5^{\left[ {5 \times \left( { - \frac{2}{5}} \right)} \right]}}$$
$$ = {\left( {0.5} \right)^{ - 2}}$$
$$ = \frac{1}{{{{\left( {0.5} \right)}^2}}}$$
$$ = \frac{1}{{0.25}}$$
$$ = 4$$

$$\frac{1}{{\sqrt 3 + \sqrt 4 }}$$
$$ = \frac{1}{{\sqrt 4 + \sqrt 3 }} \times \frac{{\sqrt 4 - \sqrt 3 }}{{\sqrt 4 - \sqrt 3 }}$$
$$ = \frac{{\sqrt 4 - \sqrt 3 }}{1}$$
$$ = \sqrt 4 - \sqrt 3 $$
$${\text{Similarly}}$$
$$\frac{1}{{\sqrt 4 + \sqrt 5 }} = \sqrt 5 - \sqrt 4 $$
$$ ⇒ \frac{1}{{\sqrt 5 + \sqrt 6 }} = \sqrt 6 - \sqrt 5 $$
$$ ⇒ \frac{1}{{\sqrt 6 + \sqrt 7 }} = \sqrt 7 - \sqrt 6 $$
$$ ⇒ \frac{1}{{\sqrt 7 + \sqrt 8 }} = \sqrt 8 - \sqrt 7 $$
$$ ⇒ \frac{1}{{\sqrt 8 + \sqrt 9 }} = \sqrt 9 - \sqrt 8 $$
$${\text{Now put values}}$$
$$ ⇒ \sqrt 4 - \sqrt 3 + \sqrt 5 - \sqrt 4 + \sqrt 6 - \sqrt 5 + \sqrt 7 - \sqrt 6 + \sqrt 8 - \sqrt 7 + \sqrt 9 - \sqrt 8 $$
$$ ⇒ \sqrt 9 - \sqrt 3 $$
$$ ⇒ 3 - \sqrt 3 $$