Solve frac{sqrt{2^{9}}}{sqrt[3]{2^9}}+frac{sqrt{2^35^3}}{sqrt[3]{2^9}}

$$\frac { \sqrt { 2 ^ { 9 } } } { \sqrt[ 3 ] { 2 ^ { 9 } } } + \frac { \sqrt { 2 ^ { 3 } 5 ^ { 3 } } } { \sqrt[ 3 ] { 2 ^ { 9 } } }$$

Evaluate

$$\frac{5\sqrt{10}}{4}+2\sqrt{2}\approx 6.7812742$$

Solution Steps

Calculate 2 to the power of 9 and get 512.

$$\frac{\sqrt{512}}{\sqrt[3]{2^{9}}}+\frac{\sqrt{2^{3}\times 5^{3}}}{\sqrt[3]{2^{9}}}$$

Factor 512=16^{2}\times 2. Rewrite the square root of the product \sqrt{16^{2}\times 2} as the product of square roots \sqrt{16^{2}}\sqrt{2}. Take the square root of 16^{2}.

$$\frac{16\sqrt{2}}{\sqrt[3]{2^{9}}}+\frac{\sqrt{2^{3}\times 5^{3}}}{\sqrt[3]{2^{9}}}$$

Calculate 2 to the power of 9 and get 512.

$$\frac{16\sqrt{2}}{\sqrt[3]{512}}+\frac{\sqrt{2^{3}\times 5^{3}}}{\sqrt[3]{2^{9}}}$$

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

$$\frac{16\sqrt{2}}{8}+\frac{\sqrt{2^{3}\times 5^{3}}}{\sqrt[3]{2^{9}}}$$

Divide 16\sqrt{2} by 8 to get 2\sqrt{2}.

$$2\sqrt{2}+\frac{\sqrt{2^{3}\times 5^{3}}}{\sqrt[3]{2^{9}}}$$

Calculate 2 to the power of 3 and get 8.

$$2\sqrt{2}+\frac{\sqrt{8\times 5^{3}}}{\sqrt[3]{2^{9}}}$$

Calculate 5 to the power of 3 and get 125.

$$2\sqrt{2}+\frac{\sqrt{8\times 125}}{\sqrt[3]{2^{9}}}$$

Multiply 8 and 125 to get 1000.

$$2\sqrt{2}+\frac{\sqrt{1000}}{\sqrt[3]{2^{9}}}$$

Factor 1000=10^{2}\times 10. Rewrite the square root of the product \sqrt{10^{2}\times 10} as the product of square roots \sqrt{10^{2}}\sqrt{10}. Take the square root of 10^{2}.

$$2\sqrt{2}+\frac{10\sqrt{10}}{\sqrt[3]{2^{9}}}$$

Calculate 2 to the power of 9 and get 512.

$$2\sqrt{2}+\frac{10\sqrt{10}}{\sqrt[3]{512}}$$

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

$$2\sqrt{2}+\frac{10\sqrt{10}}{8}$$

Divide 10\sqrt{10} by 8 to get \frac{5}{4}\sqrt{10}.

$$2\sqrt{2}+\frac{5}{4}\sqrt{10}$$

Factor

$$\frac{5 \sqrt{10} + 8 \sqrt{2}}{4} = 6.781274199956664$$