Solve sqrt{8}+sqrt{32}*sqrt{2}-6sqrt{1/2}+sqrt[3]{27}

$$\sqrt{ 8 } + \sqrt{ 32 } \times \sqrt{ 2 } -6 \sqrt{ \frac{ 1 }{ 2 } } + \sqrt[3]{ 27 }$$

Evaluate

$$11-\sqrt{2}\approx 9.585786438$$

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}.

$$2\sqrt{2}+\sqrt{32}\sqrt{2}-6\sqrt{\frac{1}{2}}+\sqrt[3]{27}$$

Factor 32=2\times 16. Rewrite the square root of the product \sqrt{2\times 16} as the product of square roots \sqrt{2}\sqrt{16}.

$$2\sqrt{2}+\sqrt{2}\sqrt{16}\sqrt{2}-6\sqrt{\frac{1}{2}}+\sqrt[3]{27}$$

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

$$2\sqrt{2}+2\sqrt{16}-6\sqrt{\frac{1}{2}}+\sqrt[3]{27}$$

Calculate the square root of 16 and get 4.

$$2\sqrt{2}+2\times 4-6\sqrt{\frac{1}{2}}+\sqrt[3]{27}$$

Multiply 2 and 4 to get 8.

$$2\sqrt{2}+8-6\sqrt{\frac{1}{2}}+\sqrt[3]{27}$$

Rewrite the square root of the division \sqrt{\frac{1}{2}} as the division of square roots \frac{\sqrt{1}}{\sqrt{2}}.

$$2\sqrt{2}+8-6\times \frac{\sqrt{1}}{\sqrt{2}}+\sqrt[3]{27}$$

Calculate the square root of 1 and get 1.

$$2\sqrt{2}+8-6\times \frac{1}{\sqrt{2}}+\sqrt[3]{27}$$

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$2\sqrt{2}+8-6\times \frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}+\sqrt[3]{27}$$

The square of \sqrt{2} is 2.

$$2\sqrt{2}+8-6\times \frac{\sqrt{2}}{2}+\sqrt[3]{27}$$

Cancel out 2, the greatest common factor in 6 and 2.

$$2\sqrt{2}+8-3\sqrt{2}+\sqrt[3]{27}$$

Combine 2\sqrt{2} and -3\sqrt{2} to get -\sqrt{2}.

$$-\sqrt{2}+8+\sqrt[3]{27}$$

Calculate \sqrt[3]{27} and get 3.

$$-\sqrt{2}+8+3$$

Add 8 and 3 to get 11.

$$-\sqrt{2}+11$$