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

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

Evaluate

$$14-25\sqrt{14}\approx -79.541434669$$

Solution Steps

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

$$21\left(\sqrt{2}\right)^{2}+3\sqrt{2}\sqrt{7}-28\sqrt{7}\sqrt{2}-4\left(\sqrt{7}\right)^{2}$$

The square of \sqrt{2} is 2.

$$21\times 2+3\sqrt{2}\sqrt{7}-28\sqrt{7}\sqrt{2}-4\left(\sqrt{7}\right)^{2}$$

Multiply 21 and 2 to get 42.

$$42+3\sqrt{2}\sqrt{7}-28\sqrt{7}\sqrt{2}-4\left(\sqrt{7}\right)^{2}$$

To multiply \sqrt{2} and \sqrt{7}, multiply the numbers under the square root.

$$42+3\sqrt{14}-28\sqrt{7}\sqrt{2}-4\left(\sqrt{7}\right)^{2}$$

To multiply \sqrt{7} and \sqrt{2}, multiply the numbers under the square root.

$$42+3\sqrt{14}-28\sqrt{14}-4\left(\sqrt{7}\right)^{2}$$

Combine 3\sqrt{14} and -28\sqrt{14} to get -25\sqrt{14}.

$$42-25\sqrt{14}-4\left(\sqrt{7}\right)^{2}$$

The square of \sqrt{7} is 7.

$$42-25\sqrt{14}-4\times 7$$

Multiply -4 and 7 to get -28.

$$42-25\sqrt{14}-28$$

Subtract 28 from 42 to get 14.

$$14-25\sqrt{14}$$

Factor

$$\sqrt{14} {(\sqrt{14} - 25)} = -79.541434669$$