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

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

Evaluate

$$\sqrt{2}\left(\sqrt{6}-3\right)\approx -0.778539072$$

Solution Steps

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

$$4\sqrt{3}\sqrt{6}+6\sqrt{6}\sqrt{2}-10\sqrt{3}-15\sqrt{2}$$

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}.

$$4\sqrt{3}\sqrt{3}\sqrt{2}+6\sqrt{6}\sqrt{2}-10\sqrt{3}-15\sqrt{2}$$

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

$$4\times 3\sqrt{2}+6\sqrt{6}\sqrt{2}-10\sqrt{3}-15\sqrt{2}$$

Multiply 4 and 3 to get 12.

$$12\sqrt{2}+6\sqrt{6}\sqrt{2}-10\sqrt{3}-15\sqrt{2}$$

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}.

$$12\sqrt{2}+6\sqrt{2}\sqrt{3}\sqrt{2}-10\sqrt{3}-15\sqrt{2}$$

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

$$12\sqrt{2}+6\times 2\sqrt{3}-10\sqrt{3}-15\sqrt{2}$$

Multiply 6 and 2 to get 12.

$$12\sqrt{2}+12\sqrt{3}-10\sqrt{3}-15\sqrt{2}$$

Combine 12\sqrt{3} and -10\sqrt{3} to get 2\sqrt{3}.

$$12\sqrt{2}+2\sqrt{3}-15\sqrt{2}$$

Combine 12\sqrt{2} and -15\sqrt{2} to get -3\sqrt{2}.

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

Factor

$$\sqrt{2} {(\sqrt{2} \sqrt{3} - 3)} = -0.778539072$$