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

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

Evaluate

$$-11\sqrt{10}-7\approx -41.785054262$$

Solution Steps

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

$$-3\left(\sqrt{5}\right)^{2}-12\sqrt{5}\sqrt{2}+\sqrt{2}\sqrt{5}+4\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{5} is 5.

$$-3\times 5-12\sqrt{5}\sqrt{2}+\sqrt{2}\sqrt{5}+4\left(\sqrt{2}\right)^{2}$$

Multiply -3 and 5 to get -15.

$$-15-12\sqrt{5}\sqrt{2}+\sqrt{2}\sqrt{5}+4\left(\sqrt{2}\right)^{2}$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$-15-12\sqrt{10}+\sqrt{2}\sqrt{5}+4\left(\sqrt{2}\right)^{2}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$-15-12\sqrt{10}+\sqrt{10}+4\left(\sqrt{2}\right)^{2}$$

Combine -12\sqrt{10} and \sqrt{10} to get -11\sqrt{10}.

$$-15-11\sqrt{10}+4\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$-15-11\sqrt{10}+4\times 2$$

Multiply 4 and 2 to get 8.

$$-15-11\sqrt{10}+8$$

Add -15 and 8 to get -7.

$$-7-11\sqrt{10}$$