Solve sqrt{2}(sqrt{8}+5sqrt{6})-3(sqrt{27}-2)

$$\sqrt { 2 } ( \sqrt { 8 } + 5 \sqrt { 6 } ) - 3 ( \sqrt { 27 } - 2 )$$

Evaluate

$$\sqrt{3}+10\approx 11.732050808$$

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}.

$$\sqrt{2}\left(2\sqrt{2}+5\sqrt{6}\right)-3\left(\sqrt{27}-2\right)$$

Use the distributive property to multiply \sqrt{2} by 2\sqrt{2}+5\sqrt{6}.

$$2\left(\sqrt{2}\right)^{2}+5\sqrt{2}\sqrt{6}-3\left(\sqrt{27}-2\right)$$

The square of \sqrt{2} is 2.

$$2\times 2+5\sqrt{2}\sqrt{6}-3\left(\sqrt{27}-2\right)$$

Multiply 2 and 2 to get 4.

$$4+5\sqrt{2}\sqrt{6}-3\left(\sqrt{27}-2\right)$$

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}.

$$4+5\sqrt{2}\sqrt{2}\sqrt{3}-3\left(\sqrt{27}-2\right)$$

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

$$4+5\times 2\sqrt{3}-3\left(\sqrt{27}-2\right)$$

Multiply 5 and 2 to get 10.

$$4+10\sqrt{3}-3\left(\sqrt{27}-2\right)$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$4+10\sqrt{3}-3\left(3\sqrt{3}-2\right)$$

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

$$4+10\sqrt{3}-9\sqrt{3}+6$$

Combine 10\sqrt{3} and -9\sqrt{3} to get \sqrt{3}.

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

Add 4 and 6 to get 10.

$$10+\sqrt{3}$$