Answer: Option C. -> $$\frac{{16}}{{81}}$$
$$\eqalign{
& \left( {\frac{{{9^2} \times {{18}^4}}}{{{3^{16}}}}} \right) \cr
& = \frac{{{9^2} \times {{\left( {9 \times 2} \right)}^4}}}{{{3^{16}}}} \cr
& = \frac{{{{\left( {{3^2}} \right)}^2} \times {{\left( {{3^2}} \right)}^4} \times {2^4}}}{{{3^{16}}}} \cr
& = \frac{{{3^4} \times {3^8} \times {2^4}}}{{{3^{16}}}} \cr
& = \frac{{{3^{\left( {4 + 8} \right)}} \times {2^4}}}{{{3^{16}}}} \cr
& = \frac{{{3^{12}} \times {2^4}}}{{{3^{16}}}} \cr
& = \frac{{{2^4}}}{{{3^{\left( {16 - 12} \right)}}}} \cr
& = \frac{{{2^4}}}{{{3^4}}} \cr
& = \frac{{16}}{{81}} \cr} $$

Answer: Option C. -> 1
$$\eqalign{
& \frac{{\frac{3}{{2 + \sqrt 3 }} - \frac{2}{{2 - \sqrt 3 }}}}{{2 - 5\sqrt 3 }} \cr
& = \frac{{\frac{{3\left( {2 - \sqrt 3 } \right) - 2\left( {2 + \sqrt 3 } \right)}}{{\left( {2 + \sqrt 3 \,} \right)\left( {2 - \sqrt 3 } \right)}}}}{{2 - 5\sqrt 3 }} \cr
& = \frac{{6 - 3\sqrt 3 - 4 - 2\sqrt 3 }}{{\left( {2 + \sqrt 3 } \right)\left( {2 - \sqrt 3 } \right)\left( {2 - 5\sqrt 3 } \right)}} \cr
& = \frac{{2 - 5\sqrt 3 }}{{2 - 5\sqrt 3 }} \cr
& = 1 \cr} $$

Answer: Option D. -> 39.0625
$$\eqalign{
& \frac{{{4^3} \times {5^4}}}{{{4^5}}} \cr
& = \frac{{{5^4}}}{{{4^{\left( {5 - 3} \right)}}}} \cr
& = \frac{{{5^4}}}{{{4^2}}} \cr
& = \frac{{625}}{{16}} \cr
& = 39.0625 \cr} $$

Answer: Option C. -> (729)2
38 × 34
= 3(8 + 4)
= 312
= (36)2
= (729)2

Answer: Option A. -> $$\frac{{{7^5}}}{{{6^7}}}$$
$$\eqalign{
& \frac{{343 \times 49}}{{216 \times 16 \times 81}} \cr
& = \frac{{{7^3} \times {7^2}}}{{{6^3} \times {2^4} \times {3^4}}} \cr
& = \frac{{{7^{\left( {3 + 2} \right)}}}}{{{6^3} \times {{\left( {2 \times 3} \right)}^4}}} \cr
& = \frac{{{7^5}}}{{{6^3} \times {6^4}}} \cr
& = \frac{{{7^5}}}{{{6^{\left( {3 + 4} \right)}}}} \cr
& = \frac{{{7^5}}}{{{6^7}}} \cr} $$

Answer: Option A. -> 8.484
$$\eqalign{
& \sqrt 2 = 1.414 \cr
& \Rightarrow \sqrt 8 {\text{ + 2}}\sqrt {32} - 3\sqrt {128} {\text{ + 4}}\sqrt {50} \cr
& \Rightarrow 2\sqrt 2 + 2 \times 4\sqrt 2 - 3 \times 8\sqrt 2 + 4 \times 5\sqrt 2 \cr
& \Rightarrow 2\sqrt 2 + 8\sqrt 2 - 24\sqrt 2 + 20\sqrt 2 \cr
& \Rightarrow 6\sqrt 2 \cr
& \Rightarrow 6 \times 1.414 \cr
& \Rightarrow 8.484{\text{ }} \cr} $$

Answer: Option C. -> 5
$$\eqalign{
& {\text{Let}}\,{9^3} \times {\left( {81} \right)^2} \div {\left( {27} \right)^3} = {\left( 3 \right)^x}{\text{then}} \cr
& \Rightarrow {\left( 3 \right)^x}{\text{ = }}\frac{{{{\left( {{3^2}} \right)}^3} \times {{\left( {{3^4}} \right)}^2}}}{{{{\left( {{3^3}} \right)}^3}}} \cr
& \Rightarrow {\left( 3 \right)^x} = \frac{{{3^{\left( {2 \times 3} \right)}} \times {3^{\left( {4 \times 2} \right)}}}}{{{3^{\left( {3 \times 3} \right)}}}} \cr
& \Rightarrow {\left( 3 \right)^x} = \frac{{{3^6} \times {3^8}}}{{{3^9}}} \cr
& \Rightarrow {\left( 3 \right)^x} = \frac{{{3^{\left( {6 + 8} \right)}}}}{{{3^9}}} \cr
& \Rightarrow {\left( 3 \right)^x} = \frac{{{3^{14}}}}{{{3^9}}} \cr
& \Rightarrow {\left( 3 \right)^x} = {3^{\left( {14 - 9} \right)}} \cr
& \Rightarrow {\left( 3 \right)^x} = {3^5} \cr
& \Rightarrow {\left( 3 \right)^x} = 5 \cr} $$

Answer: Option A. -> $${\left( {\frac{2}{3}} \right)^9}$$
$$\eqalign{
& \frac{{16 \times 32}}{{9 \times 27 \times 81}} \cr
& = \frac{{{2^4} \times {2^5}}}{{{3^2} \times {3^3} \times {3^4}}} \cr
& = \frac{{{2^{\left( {4 + 5} \right)}}}}{{{3^{\left( {2 + 3 + 4} \right)}}}} \cr
& = \frac{{{2^9}}}{{{3^9}}} \cr
& = {\left( {\frac{2}{3}} \right)^9} \cr} $$

Answer: Option C. -> 6
$$\eqalign{
& {\text{Let ,}} \cr
& {\left( 6 \right)^4} \div {\left( {36} \right)^3} \times 216 = {6^{\left( {x - 5} \right)}} \cr
& {\text{Then,}} \cr
& {6^{\left( {x - 5} \right)}} = {\left( 6 \right)^4} \div {\left( {{6^2}} \right)^3} \times {6^3} \cr
& \Rightarrow {6^{\left( {x - 5} \right)}} = {6^4} \div {6^{\left( {2 \times 3} \right)}} \times {6^3} \cr
& \Rightarrow {6^{\left( {x - 5} \right)}} = {6^4} \div {6^6} \times {6^3} \cr
& \Rightarrow {6^{\left( {x - 5} \right)}} = {6^{\left( {4 - 6 + 3} \right)}} \cr
& \Rightarrow {6^{\left( {x - 5} \right)}} = 6 \cr
& \Rightarrow x - 5 = 1 \cr
& \Rightarrow x = 6 \cr} $$

Answer: Option C. -> 1024
$$\eqalign{
& {\left( {256} \right)^{\frac{5}{4}}} \cr
& = {\left( {{4^4}} \right)^{\frac{5}{4}}} \cr
& = {4^{\left( {4 \times \frac{5}{4}} \right)}} \cr
& = {4^5} \cr
& = 1024 \cr} $$