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

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

Evaluate

$$\sqrt{42}-\sqrt{30}-\sqrt{35}-6\sqrt{6}-14\approx -33.609503116$$

Solution Steps

Use the distributive property to multiply \sqrt{7}+\sqrt{6} by \sqrt{7}-\sqrt{5}.

$$\left(\sqrt{7}\right)^{2}-\sqrt{7}\sqrt{5}+\sqrt{6}\sqrt{7}-\sqrt{6}\sqrt{5}-\left(\sqrt{3}+3\sqrt{2}\right)^{2}$$

The square of \sqrt{7} is 7.

$$7-\sqrt{7}\sqrt{5}+\sqrt{6}\sqrt{7}-\sqrt{6}\sqrt{5}-\left(\sqrt{3}+3\sqrt{2}\right)^{2}$$

To multiply \sqrt{7} and \sqrt{5}, multiply the numbers under the square root.

$$7-\sqrt{35}+\sqrt{6}\sqrt{7}-\sqrt{6}\sqrt{5}-\left(\sqrt{3}+3\sqrt{2}\right)^{2}$$

To multiply \sqrt{6} and \sqrt{7}, multiply the numbers under the square root.

$$7-\sqrt{35}+\sqrt{42}-\sqrt{6}\sqrt{5}-\left(\sqrt{3}+3\sqrt{2}\right)^{2}$$

To multiply \sqrt{6} and \sqrt{5}, multiply the numbers under the square root.

$$7-\sqrt{35}+\sqrt{42}-\sqrt{30}-\left(\sqrt{3}+3\sqrt{2}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{3}+3\sqrt{2}\right)^{2}.

$$7-\sqrt{35}+\sqrt{42}-\sqrt{30}-\left(\left(\sqrt{3}\right)^{2}+6\sqrt{3}\sqrt{2}+9\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{3} is 3.

$$7-\sqrt{35}+\sqrt{42}-\sqrt{30}-\left(3+6\sqrt{3}\sqrt{2}+9\left(\sqrt{2}\right)^{2}\right)$$

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

$$7-\sqrt{35}+\sqrt{42}-\sqrt{30}-\left(3+6\sqrt{6}+9\left(\sqrt{2}\right)^{2}\right)$$

The square of \sqrt{2} is 2.

$$7-\sqrt{35}+\sqrt{42}-\sqrt{30}-\left(3+6\sqrt{6}+9\times 2\right)$$

Multiply 9 and 2 to get 18.

$$7-\sqrt{35}+\sqrt{42}-\sqrt{30}-\left(3+6\sqrt{6}+18\right)$$

Add 3 and 18 to get 21.

$$7-\sqrt{35}+\sqrt{42}-\sqrt{30}-\left(21+6\sqrt{6}\right)$$

To find the opposite of 21+6\sqrt{6}, find the opposite of each term.

$$7-\sqrt{35}+\sqrt{42}-\sqrt{30}-21-6\sqrt{6}$$

Subtract 21 from 7 to get -14.

$$-14-\sqrt{35}+\sqrt{42}-\sqrt{30}-6\sqrt{6}$$