Solve (sqrt{5}+sqrt{6}+sqrt{7})(sqrt{5}+sqrt{6}-sqrt{7})(sqrt{5}-sqrt{6})

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

Evaluate

$$6\sqrt{6}-8\sqrt{5}\approx -3.191605363$$

Solution Steps

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

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

The square of \sqrt{5} is 5.

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

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

$$\left(5+\sqrt{30}-\sqrt{5}\sqrt{7}+\sqrt{6}\sqrt{5}+\left(\sqrt{6}\right)^{2}-\sqrt{6}\sqrt{7}+\sqrt{7}\sqrt{5}+\sqrt{7}\sqrt{6}-\left(\sqrt{7}\right)^{2}\right)\left(\sqrt{5}-\sqrt{6}\right)$$

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

$$\left(5+\sqrt{30}-\sqrt{35}+\sqrt{6}\sqrt{5}+\left(\sqrt{6}\right)^{2}-\sqrt{6}\sqrt{7}+\sqrt{7}\sqrt{5}+\sqrt{7}\sqrt{6}-\left(\sqrt{7}\right)^{2}\right)\left(\sqrt{5}-\sqrt{6}\right)$$

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

$$\left(5+\sqrt{30}-\sqrt{35}+\sqrt{30}+\left(\sqrt{6}\right)^{2}-\sqrt{6}\sqrt{7}+\sqrt{7}\sqrt{5}+\sqrt{7}\sqrt{6}-\left(\sqrt{7}\right)^{2}\right)\left(\sqrt{5}-\sqrt{6}\right)$$

Combine \sqrt{30} and \sqrt{30} to get 2\sqrt{30}.

$$\left(5+2\sqrt{30}-\sqrt{35}+\left(\sqrt{6}\right)^{2}-\sqrt{6}\sqrt{7}+\sqrt{7}\sqrt{5}+\sqrt{7}\sqrt{6}-\left(\sqrt{7}\right)^{2}\right)\left(\sqrt{5}-\sqrt{6}\right)$$

The square of \sqrt{6} is 6.

$$\left(5+2\sqrt{30}-\sqrt{35}+6-\sqrt{6}\sqrt{7}+\sqrt{7}\sqrt{5}+\sqrt{7}\sqrt{6}-\left(\sqrt{7}\right)^{2}\right)\left(\sqrt{5}-\sqrt{6}\right)$$

Add 5 and 6 to get 11.

$$\left(11+2\sqrt{30}-\sqrt{35}-\sqrt{6}\sqrt{7}+\sqrt{7}\sqrt{5}+\sqrt{7}\sqrt{6}-\left(\sqrt{7}\right)^{2}\right)\left(\sqrt{5}-\sqrt{6}\right)$$

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

$$\left(11+2\sqrt{30}-\sqrt{35}-\sqrt{42}+\sqrt{7}\sqrt{5}+\sqrt{7}\sqrt{6}-\left(\sqrt{7}\right)^{2}\right)\left(\sqrt{5}-\sqrt{6}\right)$$

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

$$\left(11+2\sqrt{30}-\sqrt{35}-\sqrt{42}+\sqrt{35}+\sqrt{7}\sqrt{6}-\left(\sqrt{7}\right)^{2}\right)\left(\sqrt{5}-\sqrt{6}\right)$$

Combine -\sqrt{35} and \sqrt{35} to get 0.

$$\left(11+2\sqrt{30}-\sqrt{42}+\sqrt{7}\sqrt{6}-\left(\sqrt{7}\right)^{2}\right)\left(\sqrt{5}-\sqrt{6}\right)$$

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

$$\left(11+2\sqrt{30}-\sqrt{42}+\sqrt{42}-\left(\sqrt{7}\right)^{2}\right)\left(\sqrt{5}-\sqrt{6}\right)$$

Combine -\sqrt{42} and \sqrt{42} to get 0.

$$\left(11+2\sqrt{30}-\left(\sqrt{7}\right)^{2}\right)\left(\sqrt{5}-\sqrt{6}\right)$$

The square of \sqrt{7} is 7.

$$\left(11+2\sqrt{30}-7\right)\left(\sqrt{5}-\sqrt{6}\right)$$

Subtract 7 from 11 to get 4.

$$\left(4+2\sqrt{30}\right)\left(\sqrt{5}-\sqrt{6}\right)$$

Apply the distributive property by multiplying each term of 4+2\sqrt{30} by each term of \sqrt{5}-\sqrt{6}.

$$4\sqrt{5}-4\sqrt{6}+2\sqrt{30}\sqrt{5}-2\sqrt{6}\sqrt{30}$$

Factor 30=5\times 6. Rewrite the square root of the product \sqrt{5\times 6} as the product of square roots \sqrt{5}\sqrt{6}.

$$4\sqrt{5}-4\sqrt{6}+2\sqrt{5}\sqrt{6}\sqrt{5}-2\sqrt{6}\sqrt{30}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$4\sqrt{5}-4\sqrt{6}+2\times 5\sqrt{6}-2\sqrt{6}\sqrt{30}$$

Multiply 2 and 5 to get 10.

$$4\sqrt{5}-4\sqrt{6}+10\sqrt{6}-2\sqrt{6}\sqrt{30}$$

Combine -4\sqrt{6} and 10\sqrt{6} to get 6\sqrt{6}.

$$4\sqrt{5}+6\sqrt{6}-2\sqrt{6}\sqrt{30}$$

Factor 30=6\times 5. Rewrite the square root of the product \sqrt{6\times 5} as the product of square roots \sqrt{6}\sqrt{5}.

$$4\sqrt{5}+6\sqrt{6}-2\sqrt{6}\sqrt{6}\sqrt{5}$$

Multiply \sqrt{6} and \sqrt{6} to get 6.

$$4\sqrt{5}+6\sqrt{6}-2\times 6\sqrt{5}$$

Multiply -2 and 6 to get -12.

$$4\sqrt{5}+6\sqrt{6}-12\sqrt{5}$$

Combine 4\sqrt{5} and -12\sqrt{5} to get -8\sqrt{5}.

$$-8\sqrt{5}+6\sqrt{6}$$