d. 2^(3)/(2)times sqrt (2) e. (5sqrt (24)div 2sqrt (50))/(3sqrt (3)) f. sqrt [3](0.001)+sqrt [4](0.0081) g (5^-1)^3times 5^(5)/(4)times sqrt (25)times (sqrt [3](125))^3 h (sqrt (0.64))times (sqrt [3]((1)/(64)))^2times 32^(1)/(5) i. 3^(1)/(4)times 27^(1)/(4) j. sqrt (5+2sqrt (6))+sqrt (8-2sqrt (15))

Let's go through each expression step by step:<br /><br />d. \(2^{\frac{3}{2}} \times \sqrt{2}\)<br /><br />First, express \(\sqrt{2}\) as \(2^{\frac{1}{2}}\):<br />\[ 2^{\frac{3}{2}} \times 2^{\frac{1}{2}} = 2^{\frac{3}{2} + \frac{1}{2}} = 2^2 = 4 \]<br /><br />e. \(\frac{5\sqrt{24} \div 2\sqrt{50}}{3\sqrt{3}}\)<br /><br />Simplify the numerator:<br />\[ 5\sqrt{24} \div 2\sqrt{50} = \frac{5\sqrt{24}}{2\sqrt{50}} = \frac{5\sqrt{4 \times 6}}{2\sqrt{25 \times 2}} = \frac{5 \times 2\sqrt{6}}{2 \times 5\sqrt{2}} = \frac{10\sqrt{6}}{10\sqrt{2}} = \frac{\sqrt{6}}{\sqrt{2}} = \sqrt{3} \]<br /><br />Now, divide by the denominator:<br />\[ \frac{\sqrt{3}}{3\sqrt{3}} = \frac{1}{3} \]<br /><br />f. \(\sqrt[3]{0.001} + \sqrt[4]{0.0081}\)<br /><br />Convert to exponents:<br />\[ \sqrt[3]{0.001} = (0.001)^{\frac{1}{3}} = (10^{-3})^{\frac{1}{3}} = 10^{-1} = 0.1 \]<br />\[ \sqrt[4]{0.0081} = (0.0081)^{\frac{1}{4}} = (81 \times 10^{-4})^{\frac{1}{4}} = 81^{\frac{1}{4}} \times 10^{-1} = 3 \times 10^{-1} = 0.3 \]<br /><br />Add them together:<br />\[ 0.1 + 0.3 = 0.4 \]<br /><br />g. \((5^{-1})^3 \times 5^{\frac{5}{4}} \times \sqrt{25} \times (\sqrt[3]{125})^3\)<br /><br />Simplify each term:<br />\[ (5^{-1})^3 = 5^{-3} \]<br />\[ 5^{\frac{5}{4}} \]<br />\[ \sqrt{25} = 5 \]<br />\[ (\sqrt[3]{125})^3 = 125^{\frac{1}{3} \times 3} = 125 \]<br /><br />Combine:<br />\[ 5^{-3} \times 5^{\frac{5}{4}} \times 5 \times 125 \]<br /><br />Convert 125 to a power of 5:<br />\[ 125 = 5^3 \]<br /><br />Combine all terms:<br />\[ 5^{-3 + \frac{5}{4} + 1 + 3} = 5^{\frac{5}{4}} \]<br /><br />h. \((\sqrt{0.64}) \times (\sqrt[3]{\frac{1}{64}})^2 \times 32^{\frac{1}{5}}\)<br /><br />Simplify each term:<br />\[ \sqrt{0.64} = 0.8 \]<br />\[ \sqrt[3]{\frac{1}{64}} = \left(\frac{1}{64}\right)^{\frac{1}{3}} = \frac{1}{4} \]<br />\[ 32^{\frac{1}{5}} = 2 \]<br /><br />Combine:<br />\[ 0.8 \times \left(\frac{1}{4}\right)^2 \times 2 = 0.8 \times \frac{1}{16} \times 2 = 0.8 \times \frac{1}{8} = 0.1 \]<br /><br />i. \(3^{\frac{1}{4}} \times 27^{\frac{1}{4}}\)<br /><br />Convert 27 to a power of 3:<br />\[ 27 = 3^3 \]<br />\[ 27^{\frac{1}{4}} = (3^3)^{\frac{1}{4}} = 3^{\frac{3}{4}} \]<br /><br />Combine:<br />\[ 3^{\frac{1}{4}} \times 3^{\frac{3}{4}} = 3^{\frac{1}{4} + \frac{3}{4}} = 3^1 = 3 \]<br /><br />j. \(\sqrt{5 + 2\sqrt{6}} + \sqrt{8 - 2\sqrt{15