Solve (1+sqrt{2}+sqrt{3})(1+sqrt{8})

$$( 1 + \sqrt { 2 } + \sqrt { 3 } ) ( 1 + \sqrt { 8 } )$$

Evaluate

$$\sqrt{3}+2\sqrt{6}+3\sqrt{2}+5\approx 15.87367098$$

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}.

$$\left(1+\sqrt{2}+\sqrt{3}\right)\left(1+2\sqrt{2}\right)$$

Apply the distributive property by multiplying each term of 1+\sqrt{2}+\sqrt{3} by each term of 1+2\sqrt{2}.

$$1+2\sqrt{2}+\sqrt{2}+2\left(\sqrt{2}\right)^{2}+\sqrt{3}+2\sqrt{3}\sqrt{2}$$

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

$$1+3\sqrt{2}+2\left(\sqrt{2}\right)^{2}+\sqrt{3}+2\sqrt{3}\sqrt{2}$$

The square of \sqrt{2} is 2.

$$1+3\sqrt{2}+2\times 2+\sqrt{3}+2\sqrt{3}\sqrt{2}$$

Multiply 2 and 2 to get 4.

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

Add 1 and 4 to get 5.

$$5+3\sqrt{2}+\sqrt{3}+2\sqrt{3}\sqrt{2}$$

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

$$5+3\sqrt{2}+\sqrt{3}+2\sqrt{6}$$