Solve (7sqrt{2}-5sqrt{6}-3sqrt{8}+4sqrt{20})*3sqrt{2}

$$( 7 \sqrt { 2 } - 5 \sqrt { 6 } - 3 \sqrt { 8 } + 4 \sqrt { 20 } ) \cdot 3 \sqrt { 2 }$$

Evaluate

$$24\sqrt{10}+6-30\sqrt{3}\approx 29.933139617$$

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}.

$$3\left(7\sqrt{2}-5\sqrt{6}-3\times 2\sqrt{2}+4\sqrt{20}\right)\sqrt{2}$$

Multiply -3 and 2 to get -6.

$$3\left(7\sqrt{2}-5\sqrt{6}-6\sqrt{2}+4\sqrt{20}\right)\sqrt{2}$$

Combine 7\sqrt{2} and -6\sqrt{2} to get \sqrt{2}.

$$3\left(\sqrt{2}-5\sqrt{6}+4\sqrt{20}\right)\sqrt{2}$$

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}.

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

Multiply 4 and 2 to get 8.

$$3\left(\sqrt{2}-5\sqrt{6}+8\sqrt{5}\right)\sqrt{2}$$

Use the distributive property to multiply 3 by \sqrt{2}-5\sqrt{6}+8\sqrt{5}.

$$\left(3\sqrt{2}-15\sqrt{6}+24\sqrt{5}\right)\sqrt{2}$$

Use the distributive property to multiply 3\sqrt{2}-15\sqrt{6}+24\sqrt{5} by \sqrt{2}.

$$3\left(\sqrt{2}\right)^{2}-15\sqrt{6}\sqrt{2}+24\sqrt{5}\sqrt{2}$$

The square of \sqrt{2} is 2.

$$3\times 2-15\sqrt{6}\sqrt{2}+24\sqrt{5}\sqrt{2}$$

Multiply 3 and 2 to get 6.

$$6-15\sqrt{6}\sqrt{2}+24\sqrt{5}\sqrt{2}$$

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

$$6-15\sqrt{2}\sqrt{3}\sqrt{2}+24\sqrt{5}\sqrt{2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$6-15\times 2\sqrt{3}+24\sqrt{5}\sqrt{2}$$

Multiply -15 and 2 to get -30.

$$6-30\sqrt{3}+24\sqrt{5}\sqrt{2}$$

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

$$6-30\sqrt{3}+24\sqrt{10}$$