Solve o(sqrt[3]{sqrt[3]{2^9}})^5(sqrt[6]{sqrt[3]{2^9}})^5

$$o ( \sqrt[ 3 ] { \sqrt[ 3 ] { 2 ^ { 9 } } } ) ^ { 5 } ( \sqrt[ 6 ] { \sqrt[ 3 ] { 2 ^ { 9 } } } ) ^ { 5 }$$

Evaluate

$$128\sqrt{2}o$$

Solution Steps

Calculate 2 to the power of 9 and get 512.

$$o\left(\sqrt[3]{\sqrt[3]{512}}\right)^{5}\left(\sqrt[6]{\sqrt[3]{2^{9}}}\right)^{5}$$

Calculate \sqrt[3]{512} and get 8.

$$o\left(\sqrt[3]{8}\right)^{5}\left(\sqrt[6]{\sqrt[3]{2^{9}}}\right)^{5}$$

Calculate \sqrt[3]{8} and get 2.

$$o\times 2^{5}\left(\sqrt[6]{\sqrt[3]{2^{9}}}\right)^{5}$$

Calculate 2 to the power of 5 and get 32.

$$o\times 32\left(\sqrt[6]{\sqrt[3]{2^{9}}}\right)^{5}$$

Calculate 2 to the power of 9 and get 512.

$$o\times 32\left(\sqrt[6]{\sqrt[3]{512}}\right)^{5}$$

Calculate \sqrt[3]{512} and get 8.

$$o\times 32\left(\sqrt[6]{8}\right)^{5}$$

Rewrite \sqrt[6]{8} as \sqrt[6]{2^{3}}. Convert from radical to exponential form and cancel out 3 in the exponent. Convert back to radical form.

$$\sqrt[6]{8}=\sqrt[6]{2^{3}}=2^{\frac{3}{6}}=2^{\frac{1}{2}}=\sqrt{2}$$

Insert the obtained value back in the expression.

$$o\times 32\left(\sqrt{2}\right)^{5}$$

Differentiate w.r.t. o

$$128 \sqrt{2} = 181.019335984$$