Solve sqrt[3]{10sqrt{8}*sqrt{50}*sqrt[3]{250*2^2}}+sqrt[3]{-2}

$$\sqrt[ 3 ] { 10 \sqrt { 8 } \cdot \sqrt { 50 } \cdot \sqrt[ 3 ] { 250 \cdot 2 ^ { 2 } } } + \sqrt[ 3 ] { - 2 }$$

Evaluate

$$9\sqrt[3]{2}\approx 11.339289449$$

Solution Steps

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\sqrt[3]{10\times 2\sqrt{2}\sqrt{50}\sqrt[3]{250\times 2^{2}}}+\sqrt[3]{-2}$$

Multiply 10 and 2 to get 20.

$$\sqrt[3]{20\sqrt{2}\sqrt{50}\sqrt[3]{250\times 2^{2}}}+\sqrt[3]{-2}$$

Factor 50=5^{2}\times 2. Rewrite the square root of the product \sqrt{5^{2}\times 2} as the product of square roots \sqrt{5^{2}}\sqrt{2}. Take the square root of 5^{2}.

$$\sqrt[3]{20\sqrt{2}\times 5\sqrt{2}\sqrt[3]{250\times 2^{2}}}+\sqrt[3]{-2}$$

Multiply 20 and 5 to get 100.

$$\sqrt[3]{100\sqrt{2}\sqrt{2}\sqrt[3]{250\times 2^{2}}}+\sqrt[3]{-2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\sqrt[3]{100\times 2\sqrt[3]{250\times 2^{2}}}+\sqrt[3]{-2}$$

Calculate 2 to the power of 2 and get 4.

$$\sqrt[3]{100\times 2\sqrt[3]{250\times 4}}+\sqrt[3]{-2}$$

Multiply 250 and 4 to get 1000.

$$\sqrt[3]{100\times 2\sqrt[3]{1000}}+\sqrt[3]{-2}$$

Calculate \sqrt[3]{1000} and get 10.

$$\sqrt[3]{100\times 2\times 10}+\sqrt[3]{-2}$$

Multiply 100 and 10 to get 1000.

$$\sqrt[3]{1000\times 2}+\sqrt[3]{-2}$$

Multiply 1000 and 2 to get 2000.

$$\sqrt[3]{2000}+\sqrt[3]{-2}$$