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

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

Evaluate

$$8-\sqrt{15}\approx 4.127016654$$

Solution Steps

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

$$6\left(\sqrt{3}\right)^{2}-4\sqrt{3}\sqrt{5}+3\sqrt{5}\sqrt{3}-2\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{3} is 3.

$$6\times 3-4\sqrt{3}\sqrt{5}+3\sqrt{5}\sqrt{3}-2\left(\sqrt{5}\right)^{2}$$

Multiply 6 and 3 to get 18.

$$18-4\sqrt{3}\sqrt{5}+3\sqrt{5}\sqrt{3}-2\left(\sqrt{5}\right)^{2}$$

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$18-4\sqrt{15}+3\sqrt{5}\sqrt{3}-2\left(\sqrt{5}\right)^{2}$$

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$18-4\sqrt{15}+3\sqrt{15}-2\left(\sqrt{5}\right)^{2}$$

Combine -4\sqrt{15} and 3\sqrt{15} to get -\sqrt{15}.

$$18-\sqrt{15}-2\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$18-\sqrt{15}-2\times 5$$

Multiply -2 and 5 to get -10.

$$18-\sqrt{15}-10$$

Subtract 10 from 18 to get 8.

$$8-\sqrt{15}$$