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

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

Evaluate

$$-9\sqrt{2}-24\approx -36.727922061$$

Solution Steps

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

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

The square of \sqrt{3} is 3.

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

Multiply 2 and 3 to get 6.

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

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}.

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

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

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

Multiply -5 and 3 to get -15.

$$6-15\sqrt{2}+2\sqrt{6}\sqrt{3}-5\left(\sqrt{6}\right)^{2}$$

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}.

$$6-15\sqrt{2}+2\sqrt{3}\sqrt{2}\sqrt{3}-5\left(\sqrt{6}\right)^{2}$$

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

$$6-15\sqrt{2}+2\times 3\sqrt{2}-5\left(\sqrt{6}\right)^{2}$$

Multiply 2 and 3 to get 6.

$$6-15\sqrt{2}+6\sqrt{2}-5\left(\sqrt{6}\right)^{2}$$

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

$$6-9\sqrt{2}-5\left(\sqrt{6}\right)^{2}$$

The square of \sqrt{6} is 6.

$$6-9\sqrt{2}-5\times 6$$

Multiply -5 and 6 to get -30.

$$6-9\sqrt{2}-30$$

Subtract 30 from 6 to get -24.

$$-24-9\sqrt{2}$$