Solve 3sqrt{2}*(3sqrt{48}-2sqrt{12}-4sqrt{1/8})

$$3 \sqrt { 2 } \times ( 3 \sqrt { 48 } - 2 \sqrt { 12 } - 4 \sqrt { \frac { 1 } { 8 } } )$$

Evaluate

$$24\sqrt{6}-6\approx 52.787753827$$

Solution Steps

Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.

$$3\sqrt{2}\left(3\times 4\sqrt{3}-2\sqrt{12}-4\sqrt{\frac{1}{8}}\right)$$

Multiply 3 and 4 to get 12.

$$3\sqrt{2}\left(12\sqrt{3}-2\sqrt{12}-4\sqrt{\frac{1}{8}}\right)$$

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}.

$$3\sqrt{2}\left(12\sqrt{3}-2\times 2\sqrt{3}-4\sqrt{\frac{1}{8}}\right)$$

Multiply -2 and 2 to get -4.

$$3\sqrt{2}\left(12\sqrt{3}-4\sqrt{3}-4\sqrt{\frac{1}{8}}\right)$$

Combine 12\sqrt{3} and -4\sqrt{3} to get 8\sqrt{3}.

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

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

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

Calculate the square root of 1 and get 1.

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

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}.

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

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

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

The square of \sqrt{2} is 2.

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

Multiply 2 and 2 to get 4.

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

Cancel out 4 and 4.

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

Use the distributive property to multiply 3\sqrt{2} by 8\sqrt{3}-\sqrt{2}.

$$24\sqrt{3}\sqrt{2}-3\left(\sqrt{2}\right)^{2}$$

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

$$24\sqrt{6}-3\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$24\sqrt{6}-3\times 2$$

Multiply -3 and 2 to get -6.

$$24\sqrt{6}-6$$