Solve sqrt{2}(2sqrt{8}+3sqrt{12})

$$\sqrt { 2 } ( 2 \sqrt { 8 } + 3 \sqrt { 12 } )$$

Evaluate

$$6\sqrt{6}+8\approx 22.696938457$$

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}.

$$\sqrt{2}\left(2\times 2\sqrt{2}+3\sqrt{12}\right)$$

Multiply 2 and 2 to get 4.

$$\sqrt{2}\left(4\sqrt{2}+3\sqrt{12}\right)$$

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}.

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

Multiply 3 and 2 to get 6.

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

Use the distributive property to multiply \sqrt{2} by 4\sqrt{2}+6\sqrt{3}.

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

The square of \sqrt{2} is 2.

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

Multiply 4 and 2 to get 8.

$$8+6\sqrt{2}\sqrt{3}$$

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

$$8+6\sqrt{6}$$