Solve (sqrt{3}-2sqrt{5})(4sqrt{3}+sqrt{5})(sqrt{15}-3)

$$( \sqrt { 3 } - 2 \sqrt { 5 } ) ( 4 \sqrt { 3 } + \sqrt { 5 } ) ( \sqrt { 15 } - 3 )$$

Evaluate

$$23\sqrt{15}-111\approx -21.921383037$$

Solution Steps

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

$$\left(4\left(\sqrt{3}\right)^{2}+\sqrt{3}\sqrt{5}-8\sqrt{3}\sqrt{5}-2\left(\sqrt{5}\right)^{2}\right)\left(\sqrt{15}-3\right)$$

The square of \sqrt{3} is 3.

$$\left(4\times 3+\sqrt{3}\sqrt{5}-8\sqrt{3}\sqrt{5}-2\left(\sqrt{5}\right)^{2}\right)\left(\sqrt{15}-3\right)$$

Multiply 4 and 3 to get 12.

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

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

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

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

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

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

$$\left(12-7\sqrt{15}-2\left(\sqrt{5}\right)^{2}\right)\left(\sqrt{15}-3\right)$$

The square of \sqrt{5} is 5.

$$\left(12-7\sqrt{15}-2\times 5\right)\left(\sqrt{15}-3\right)$$

Multiply -2 and 5 to get -10.

$$\left(12-7\sqrt{15}-10\right)\left(\sqrt{15}-3\right)$$

Subtract 10 from 12 to get 2.

$$\left(2-7\sqrt{15}\right)\left(\sqrt{15}-3\right)$$

Apply the distributive property by multiplying each term of 2-7\sqrt{15} by each term of \sqrt{15}-3.

$$2\sqrt{15}-6-7\left(\sqrt{15}\right)^{2}+21\sqrt{15}$$

The square of \sqrt{15} is 15.

$$2\sqrt{15}-6-7\times 15+21\sqrt{15}$$

Multiply -7 and 15 to get -105.

$$2\sqrt{15}-6-105+21\sqrt{15}$$

Subtract 105 from -6 to get -111.

$$2\sqrt{15}-111+21\sqrt{15}$$

Combine 2\sqrt{15} and 21\sqrt{15} to get 23\sqrt{15}.

$$23\sqrt{15}-111$$