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

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

Evaluate

$$3\sqrt{15}-4\approx 7.618950039$$

Solution Steps

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

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

The square of \sqrt{3} is 3.

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

Multiply 2 and 3 to get 6.

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

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

$$6+4\sqrt{15}-\sqrt{5}\sqrt{3}-2\left(\sqrt{5}\right)^{2}$$

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

$$6+4\sqrt{15}-\sqrt{15}-2\left(\sqrt{5}\right)^{2}$$

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

$$6+3\sqrt{15}-2\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$6+3\sqrt{15}-2\times 5$$

Multiply -2 and 5 to get -10.

$$6+3\sqrt{15}-10$$

Subtract 10 from 6 to get -4.

$$-4+3\sqrt{15}$$