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

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

Evaluate

$$9\sqrt{2}-4\sqrt{3}\approx 5.799718831$$

Solution Steps

Apply the distributive property by multiplying each term of 5+\sqrt{6} by each term of 3\sqrt{2}-2\sqrt{3}.

$$15\sqrt{2}-10\sqrt{3}+3\sqrt{6}\sqrt{2}-2\sqrt{6}\sqrt{3}$$

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}.

$$15\sqrt{2}-10\sqrt{3}+3\sqrt{2}\sqrt{3}\sqrt{2}-2\sqrt{6}\sqrt{3}$$

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

$$15\sqrt{2}-10\sqrt{3}+3\times 2\sqrt{3}-2\sqrt{6}\sqrt{3}$$

Multiply 3 and 2 to get 6.

$$15\sqrt{2}-10\sqrt{3}+6\sqrt{3}-2\sqrt{6}\sqrt{3}$$

Combine -10\sqrt{3} and 6\sqrt{3} to get -4\sqrt{3}.

$$15\sqrt{2}-4\sqrt{3}-2\sqrt{6}\sqrt{3}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$15\sqrt{2}-4\sqrt{3}-2\sqrt{3}\sqrt{2}\sqrt{3}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$15\sqrt{2}-4\sqrt{3}-2\times 3\sqrt{2}$$

Multiply -2 and 3 to get -6.

$$15\sqrt{2}-4\sqrt{3}-6\sqrt{2}$$

Combine 15\sqrt{2} and -6\sqrt{2} to get 9\sqrt{2}.

$$9\sqrt{2}-4\sqrt{3}$$

Factor

$$9 \sqrt{2} - 4 \sqrt{3} = 5.799718831$$