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

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

Evaluate

$$2-7\sqrt{15}\approx -25.110883423$$

Solution Steps

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

$$4\left(\sqrt{3}\right)^{2}-8\sqrt{3}\sqrt{5}+\sqrt{5}\sqrt{3}-2\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{3} is 3.

$$4\times 3-8\sqrt{3}\sqrt{5}+\sqrt{5}\sqrt{3}-2\left(\sqrt{5}\right)^{2}$$

Multiply 4 and 3 to get 12.

$$12-8\sqrt{3}\sqrt{5}+\sqrt{5}\sqrt{3}-2\left(\sqrt{5}\right)^{2}$$

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

$$12-8\sqrt{15}+\sqrt{5}\sqrt{3}-2\left(\sqrt{5}\right)^{2}$$

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

$$12-8\sqrt{15}+\sqrt{15}-2\left(\sqrt{5}\right)^{2}$$

Combine -8\sqrt{15} and \sqrt{15} to get -7\sqrt{15}.

$$12-7\sqrt{15}-2\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$12-7\sqrt{15}-2\times 5$$

Multiply -2 and 5 to get -10.

$$12-7\sqrt{15}-10$$

Subtract 10 from 12 to get 2.

$$2-7\sqrt{15}$$