Solve (5sqrt{3}+7sqrt{2})(6sqrt{3}-4sqrt{2})

$$( 5 \sqrt { 3 } + 7 \sqrt { 2 } ) ( 6 \sqrt { 3 } - 4 \sqrt { 2 } )$$

Evaluate

$$22\sqrt{6}+34\approx 87.888774341$$

Solution Steps

Apply the distributive property by multiplying each term of 5\sqrt{3}+7\sqrt{2} by each term of 6\sqrt{3}-4\sqrt{2}.

$$30\left(\sqrt{3}\right)^{2}-20\sqrt{3}\sqrt{2}+42\sqrt{3}\sqrt{2}-28\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{3} is 3.

$$30\times 3-20\sqrt{3}\sqrt{2}+42\sqrt{3}\sqrt{2}-28\left(\sqrt{2}\right)^{2}$$

Multiply 30 and 3 to get 90.

$$90-20\sqrt{3}\sqrt{2}+42\sqrt{3}\sqrt{2}-28\left(\sqrt{2}\right)^{2}$$

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

$$90-20\sqrt{6}+42\sqrt{3}\sqrt{2}-28\left(\sqrt{2}\right)^{2}$$

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

$$90-20\sqrt{6}+42\sqrt{6}-28\left(\sqrt{2}\right)^{2}$$

Combine -20\sqrt{6} and 42\sqrt{6} to get 22\sqrt{6}.

$$90+22\sqrt{6}-28\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$90+22\sqrt{6}-28\times 2$$

Multiply -28 and 2 to get -56.

$$90+22\sqrt{6}-56$$

Subtract 56 from 90 to get 34.

$$34+22\sqrt{6}$$

Factor

$$2 {(11 \sqrt{6} + 17)} = 87.888774341$$