Solve (7sqrt{5}-3sqrt{2})*(2sqrt{5}+4sqrt{2})

$$( 7 \sqrt { 5 } - 3 \sqrt { 2 } ) \times ( 2 \sqrt { 5 } + 4 \sqrt { 2 } )$$

Evaluate

$$22\sqrt{10}+46\approx 115.570108524$$

Solution Steps

Apply the distributive property by multiplying each term of 7\sqrt{5}-3\sqrt{2} by each term of 2\sqrt{5}+4\sqrt{2}.

$$14\left(\sqrt{5}\right)^{2}+28\sqrt{5}\sqrt{2}-6\sqrt{2}\sqrt{5}-12\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{5} is 5.

$$14\times 5+28\sqrt{5}\sqrt{2}-6\sqrt{2}\sqrt{5}-12\left(\sqrt{2}\right)^{2}$$

Multiply 14 and 5 to get 70.

$$70+28\sqrt{5}\sqrt{2}-6\sqrt{2}\sqrt{5}-12\left(\sqrt{2}\right)^{2}$$

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

$$70+28\sqrt{10}-6\sqrt{2}\sqrt{5}-12\left(\sqrt{2}\right)^{2}$$

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

$$70+28\sqrt{10}-6\sqrt{10}-12\left(\sqrt{2}\right)^{2}$$

Combine 28\sqrt{10} and -6\sqrt{10} to get 22\sqrt{10}.

$$70+22\sqrt{10}-12\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$70+22\sqrt{10}-12\times 2$$

Multiply -12 and 2 to get -24.

$$70+22\sqrt{10}-24$$

Subtract 24 from 70 to get 46.

$$46+22\sqrt{10}$$

Factor

$$2 {(11 \sqrt{10} + 23)} = 115.570108524$$