Solve sqrt{12}-4sqrt{1/8}*(sqrt{3}-sqrt{8})

$$\sqrt { 12 } - 4 \sqrt { \frac { 1 } { 8 } } \times ( \sqrt { 3 } - \sqrt { 8 } )$$

Evaluate

$$2\sqrt{3}+4-\sqrt{6}\approx 5.014611872$$

Solution Steps

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

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

Rewrite the square root of the division \sqrt{\frac{1}{8}} as the division of square roots \frac{\sqrt{1}}{\sqrt{8}}.

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

Calculate the square root of 1 and get 1.

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

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}.

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

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

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

The square of \sqrt{2} is 2.

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

Multiply 2 and 2 to get 4.

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

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}.

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

Cancel out 4 and 4.

$$2\sqrt{3}-\sqrt{2}\left(\sqrt{3}-2\sqrt{2}\right)$$

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

$$2\sqrt{3}-\left(\sqrt{2}\sqrt{3}-2\left(\sqrt{2}\right)^{2}\right)$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$2\sqrt{3}-\left(\sqrt{6}-2\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{2} is 2.

$$2\sqrt{3}-\left(\sqrt{6}-2\times 2\right)$$

Multiply -2 and 2 to get -4.

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

To find the opposite of \sqrt{6}-4, find the opposite of each term.

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

The opposite of -4 is 4.

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