Solve (6sqrt{15}-2sqrt{5})(sqrt{8}+2sqrt{20})

$$(6 \sqrt{ 15 } -2 \sqrt{ 5 } )( \sqrt{ 8 } +2 \sqrt{ 20 } )$$

Evaluate

$$12\sqrt{30}+120\sqrt{3}-4\sqrt{10}-40\approx 220.923693168$$

Solution Steps

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\left(6\sqrt{15}-2\sqrt{5}\right)\left(2\sqrt{2}+2\sqrt{20}\right)$$

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

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

Multiply 2 and 2 to get 4.

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

Apply the distributive property by multiplying each term of 6\sqrt{15}-2\sqrt{5} by each term of 2\sqrt{2}+4\sqrt{5}.

$$12\sqrt{15}\sqrt{2}+24\sqrt{15}\sqrt{5}-4\sqrt{5}\sqrt{2}-8\left(\sqrt{5}\right)^{2}$$

To multiply \sqrt{15} and \sqrt{2}, multiply the numbers under the square root.

$$12\sqrt{30}+24\sqrt{15}\sqrt{5}-4\sqrt{5}\sqrt{2}-8\left(\sqrt{5}\right)^{2}$$

Factor 15=5\times 3. Rewrite the square root of the product \sqrt{5\times 3} as the product of square roots \sqrt{5}\sqrt{3}.

$$12\sqrt{30}+24\sqrt{5}\sqrt{3}\sqrt{5}-4\sqrt{5}\sqrt{2}-8\left(\sqrt{5}\right)^{2}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$12\sqrt{30}+24\times 5\sqrt{3}-4\sqrt{5}\sqrt{2}-8\left(\sqrt{5}\right)^{2}$$

Multiply 24 and 5 to get 120.

$$12\sqrt{30}+120\sqrt{3}-4\sqrt{5}\sqrt{2}-8\left(\sqrt{5}\right)^{2}$$

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

$$12\sqrt{30}+120\sqrt{3}-4\sqrt{10}-8\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$12\sqrt{30}+120\sqrt{3}-4\sqrt{10}-8\times 5$$

Multiply -8 and 5 to get -40.

$$12\sqrt{30}+120\sqrt{3}-4\sqrt{10}-40$$

Factor

$$4 {(3 \sqrt{30} + 30 \sqrt{3} - \sqrt{10} - 10)} = 220.923693168$$