Solve left(sqrt{8}+sqrt{3}right)*sqrt{6}-4sqrt{1/2}

$$\left( \sqrt{ 8 } + \sqrt{ 3 } \right) \times \sqrt{ 6 } -4 \sqrt{ \frac{ 1 }{ 2 } }$$

Evaluate

$$\sqrt{2}+4\sqrt{3}\approx 8.342416793$$

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}.

$$\left(2\sqrt{2}+\sqrt{3}\right)\sqrt{6}-4\sqrt{\frac{1}{2}}$$

Use the distributive property to multiply 2\sqrt{2}+\sqrt{3} by \sqrt{6}.

$$2\sqrt{2}\sqrt{6}+\sqrt{3}\sqrt{6}-4\sqrt{\frac{1}{2}}$$

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

$$2\sqrt{2}\sqrt{2}\sqrt{3}+\sqrt{3}\sqrt{6}-4\sqrt{\frac{1}{2}}$$

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

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

Multiply 2 and 2 to get 4.

$$4\sqrt{3}+\sqrt{3}\sqrt{6}-4\sqrt{\frac{1}{2}}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$4\sqrt{3}+\sqrt{3}\sqrt{3}\sqrt{2}-4\sqrt{\frac{1}{2}}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$4\sqrt{3}+3\sqrt{2}-4\sqrt{\frac{1}{2}}$$

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

$$4\sqrt{3}+3\sqrt{2}-4\times \frac{\sqrt{1}}{\sqrt{2}}$$

Calculate the square root of 1 and get 1.

$$4\sqrt{3}+3\sqrt{2}-4\times \frac{1}{\sqrt{2}}$$

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

$$4\sqrt{3}+3\sqrt{2}-4\times \frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$4\sqrt{3}+3\sqrt{2}-4\times \frac{\sqrt{2}}{2}$$

Cancel out 2, the greatest common factor in 4 and 2.

$$4\sqrt{3}+3\sqrt{2}-2\sqrt{2}$$

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

$$4\sqrt{3}+\sqrt{2}$$