Solve (4sqrt{2}+sqrt{7})(8sqrt{2}-6sqrt{7})

$$( 4 \sqrt { 2 } + \sqrt { 7 } ) ( 8 \sqrt { 2 } - 6 \sqrt { 7 } )$$

Evaluate

$$22-16\sqrt{14}\approx -37.866518188$$

Solution Steps

Apply the distributive property by multiplying each term of 4\sqrt{2}+\sqrt{7} by each term of 8\sqrt{2}-6\sqrt{7}.

$$32\left(\sqrt{2}\right)^{2}-24\sqrt{2}\sqrt{7}+8\sqrt{7}\sqrt{2}-6\left(\sqrt{7}\right)^{2}$$

The square of \sqrt{2} is 2.

$$32\times 2-24\sqrt{2}\sqrt{7}+8\sqrt{7}\sqrt{2}-6\left(\sqrt{7}\right)^{2}$$

Multiply 32 and 2 to get 64.

$$64-24\sqrt{2}\sqrt{7}+8\sqrt{7}\sqrt{2}-6\left(\sqrt{7}\right)^{2}$$

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

$$64-24\sqrt{14}+8\sqrt{7}\sqrt{2}-6\left(\sqrt{7}\right)^{2}$$

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

$$64-24\sqrt{14}+8\sqrt{14}-6\left(\sqrt{7}\right)^{2}$$

Combine -24\sqrt{14} and 8\sqrt{14} to get -16\sqrt{14}.

$$64-16\sqrt{14}-6\left(\sqrt{7}\right)^{2}$$

The square of \sqrt{7} is 7.

$$64-16\sqrt{14}-6\times 7$$

Multiply -6 and 7 to get -42.

$$64-16\sqrt{14}-42$$

Subtract 42 from 64 to get 22.

$$22-16\sqrt{14}$$

Factor

$$2 {(11 - 8 \sqrt{14})} = -37.866518188$$