Surds And Indices

$${\text{Let ,}}$$
$${\left( 6 \right)^4} \div {\left( {36} \right)^3} \times 216 = {6^{\left( {x - 5} \right)}}$$
$${\text{Then,}}$$
$${6^{\left( {x - 5} \right)}} = {\left( 6 \right)^4} \div {\left( {{6^2}} \right)^3} \times {6^3}$$
$$ ⇒ {6^{\left( {x - 5} \right)}} = {6^4} \div {6^{\left( {2 \times 3} \right)}} \times {6^3}$$
$$ ⇒ {6^{\left( {x - 5} \right)}} = {6^4} \div {6^6} \times {6^3}$$
$$ ⇒ {6^{\left( {x - 5} \right)}} = {6^{\left( {4 - 6 + 3} \right)}}$$
$$ ⇒ {6^{\left( {x - 5} \right)}} = 6$$
$$ ⇒×- 5 = 1$$
$$ ⇒ x = 6$$

$${\left( {256} \right)^{\frac{5}{4}}}$$
$$ = {\left( {{4^4}} \right)^{\frac{5}{4}}}$$
$$ = {4^{\left( {4 \times \frac{5}{4}} \right)}}$$
$$ = {4^5}$$
$$ = 1024$$

$$\eqalign{ &×= \sqrt {2 + \sqrt {2 + \sqrt {2 + ......} } } $$
$$ ⇒ {x^2} = 2 + \sqrt {2 + \sqrt {2 + .......} } $$
$$ ⇒ {x^2} = 2 +× $$
$$ ⇒ {x^2} -×- 2 = 0$$
$$ ⇒ x\left( {x - 2} \right) + 1\left( {x - 2} \right) = 0$$
$$ ⇒ \left( {x + 1} \right)\left( {x - 2} \right) = 0$$
$$ ⇒ x = 2$$

$${\text{2 + }}\sqrt {0.09} - \root 3 \of {0.008} - 75\% \,{\text{of }}2.80$$
$$ = 2 + 0.3 - 0.2 - \left( {\frac{3}{4} \times 2.8} \right)$$
$$ = 2 + 0.3 - 0.2 - 2.10$$
$$ = 2.3 - 2.3$$
$$ = 0$$

$${\left( {3 + 2\sqrt 2 } \right)^{ - 3}} + {\left( {3 - 2\sqrt 2 } \right)^{ - 3}}$$
$$ = {\left( {\frac{1}{{3 + 2\sqrt 2 }}} \right)^3} + {\left( {\frac{1}{{3 - 2\sqrt 2 }}} \right)^3}$$
  $$ = {\left( {\frac{1}{{\left( {3 - 2\sqrt 2 } \right)}} \times \frac{{3 + 2\sqrt 2 }}{{3 + 2\sqrt 2 }}} \right)^3} + $$      $${\left( {\frac{1}{{\left( {3 + 2\sqrt 2 } \right)}} \times \frac{{3 - 2\sqrt 2 }}{{3 - 2\sqrt 2 }}} \right)^3}$$
$$ = {\left( {\frac{{3 - 2\sqrt 2 }}{{9 - 8}}} \right)^3} + {\left( {\frac{{3 + 2\sqrt 2 }}{{9 - 8}}} \right)^3}$$
$$ = {\left( {3 - 2\sqrt 2 } \right)^3} + {\left( {3 + 2\sqrt 2 } \right)^3}$$
$$a = 3 - 2\sqrt 2 $$
$$b = 3 + 2\sqrt 2 $$
$$\left[ {\because {a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} + {b^2} - ab} \right)} \right]$$
$$ = \left( {3 - 2\sqrt 2 + 3 + 2\sqrt 2 } \right)\left( {17 + 17 - 1} \right)$$
$$ = \left( 6 \right)\left( {33} \right)$$
$$ = 198$$

$${\text{let}}$$
$$a = 0.73$$
$$b = 0.27$$
$$ = \frac{{{a^3} + {b^3}}}{{{a^2} + {b^2} - ab}}$$
$$ = \frac{{\left( {a + b} \right)\left( {{a^2} + {b^2} - ab} \right)}}{{\left( {{a^2} + {b^2} - ab} \right)}}$$
$$ = \left( {a + b} \right)$$
$$ = \left( {0.73 + 0.27} \right)$$
$$ = 1$$

$$\eqalign{ &×= \frac{{\sqrt 5 + \sqrt 3 }}{{\sqrt 5 - \sqrt 3 }}$$
$$ ⇒ x = \frac{{\sqrt 5 + \sqrt 3 }}{{\sqrt 5 - \sqrt 3 }} \times \frac{{\sqrt 5 + \sqrt 3 }}{{\sqrt 5 + \sqrt 3 }}$$
$$ ⇒ x = \frac{{{{\left( {\sqrt 5 + \sqrt 3 } \right)}^2}}}{2}$$
$${\text{Similarly}}$$
$$y = \frac{{\sqrt 5 - \sqrt 3 }}{{\sqrt 5 + \sqrt 3 }}$$
$$ ⇒ y = \frac{{{{\left( {\sqrt 5 - \sqrt 3 } \right)}^2}}}{2}$$
$${\text{Now, }}x + y$$
$$ = \frac{{5 + 3 + 2\sqrt {15} + 5 + 3 - 2\sqrt {15} }}{2}$$
$$ = \frac{{16}}{2}$$
$$ = 8$$

$${\left[ {{{\left( {\root 5 \of {{x^{ - \frac{3}{5}}}} } \right)}^{ - \frac{5}{3}}}} \right]^5}$$
$$ = {\left[ {{{\left\{ {{{\left( {{x^{ - \frac{3}{5}}}} \right)}^{\frac{1}{5}}}} \right\}}^{ - \frac{5}{3}}}} \right]^5}$$
$$ = {\left[ {{{\left( {{x^{^{\left\{ {\left( { - \frac{3}{5}} \right) \times \frac{1}{5}} \right\}}}}} \right)}^{ - \frac{5}{3}}}} \right]^5}$$
$$ = {\left[ {{{\left( {{x^{ - \frac{3}{{25}}}}} \right)}^{ - \frac{5}{3}}}} \right]^5}$$
$$ = {\left[ {{x^{\left\{ {\left( { - \frac{3}{{25}}} \right) \times \left( { - \frac{5}{3}} \right)} \right\}}}} \right]^5}$$
$$ = {\left( {{x^{\frac{1}{5}}}} \right)^5}$$
$$ = {x^{\left( {\frac{1}{5} \times 5} \right)}}$$
$$ =×\cr} $$

$${\text{Let }}\frac{{{{\left(×\right)}^{\frac{2}{3}}}}}{{42}} = \frac{5}{{{{\left(×\right)}^{\frac{1}{3}}}}}$$
$${\text{Then,}}$$
$${x^{\frac{2}{3}}}.{x^{\frac{1}{3}}} = 42 \times 5$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 210$$
$$ ⇒ {x^{\left( {\frac{2}{3} + \frac{1}{3}} \right)}} = 210$$
$$ ⇒ x = 210$$

$${{\text{5}}^{\frac{1}{4}}} \times {\left( {125} \right)^{0.25}}$$
$$ = {5^{0.25}} \times {\left( {{5^3}} \right)^{0.25}}$$
$$ = {5^{0.25}} \times {5^{\left( {3 \times 0.25} \right)}}$$
$$ = {5^{0.25}} \times {5^{0.75}}$$
$$ = {5^{\left( {0.25 + 0.75} \right)}}$$
$$ = {5^1}$$
$$ = 5$$