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

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

Evaluate

$$\sqrt{10}+6\approx 9.16227766$$

Solution Steps

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

$$6\left(\sqrt{5}\right)^{2}-8\sqrt{5}\sqrt{2}+9\sqrt{2}\sqrt{5}-12\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{5} is 5.

$$6\times 5-8\sqrt{5}\sqrt{2}+9\sqrt{2}\sqrt{5}-12\left(\sqrt{2}\right)^{2}$$

Multiply 6 and 5 to get 30.

$$30-8\sqrt{5}\sqrt{2}+9\sqrt{2}\sqrt{5}-12\left(\sqrt{2}\right)^{2}$$

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

$$30-8\sqrt{10}+9\sqrt{2}\sqrt{5}-12\left(\sqrt{2}\right)^{2}$$

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

$$30-8\sqrt{10}+9\sqrt{10}-12\left(\sqrt{2}\right)^{2}$$

Combine -8\sqrt{10} and 9\sqrt{10} to get \sqrt{10}.

$$30+\sqrt{10}-12\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$30+\sqrt{10}-12\times 2$$

Multiply -12 and 2 to get -24.

$$30+\sqrt{10}-24$$

Subtract 24 from 30 to get 6.

$$6+\sqrt{10}$$

Factor

$$\sqrt{10} + 6 = 9.16227766$$