Solve (200sqrt{3}+50)*(50sqrt{2}+75sqrt{3})*(125sqrt{2}-18)div2

$$(200 \sqrt{ 3 } +50) \times (50 \sqrt{ 2 } +75 \sqrt{ 3 } ) \times (125 \sqrt{ 2 } -18) \div 2$$

Evaluate

$$144375\sqrt{6}+1216250\sqrt{3}+2790000\sqrt{2}-92500\approx 6313407.715340903$$

Solution Steps

Apply the distributive property by multiplying each term of 200\sqrt{3}+50 by each term of 50\sqrt{2}+75\sqrt{3}.

$$\frac{\left(10000\sqrt{3}\sqrt{2}+15000\left(\sqrt{3}\right)^{2}+2500\sqrt{2}+3750\sqrt{3}\right)\left(125\sqrt{2}-18\right)}{2}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{\left(10000\sqrt{6}+15000\left(\sqrt{3}\right)^{2}+2500\sqrt{2}+3750\sqrt{3}\right)\left(125\sqrt{2}-18\right)}{2}$$

The square of \sqrt{3} is 3.

$$\frac{\left(10000\sqrt{6}+15000\times 3+2500\sqrt{2}+3750\sqrt{3}\right)\left(125\sqrt{2}-18\right)}{2}$$

Multiply 15000 and 3 to get 45000.

$$\frac{\left(10000\sqrt{6}+45000+2500\sqrt{2}+3750\sqrt{3}\right)\left(125\sqrt{2}-18\right)}{2}$$

Apply the distributive property by multiplying each term of 10000\sqrt{6}+45000+2500\sqrt{2}+3750\sqrt{3} by each term of 125\sqrt{2}-18.

$$\frac{1250000\sqrt{6}\sqrt{2}-180000\sqrt{6}+5625000\sqrt{2}-810000+312500\left(\sqrt{2}\right)^{2}-45000\sqrt{2}+468750\sqrt{3}\sqrt{2}-67500\sqrt{3}}{2}$$

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}.

$$\frac{1250000\sqrt{2}\sqrt{3}\sqrt{2}-180000\sqrt{6}+5625000\sqrt{2}-810000+312500\left(\sqrt{2}\right)^{2}-45000\sqrt{2}+468750\sqrt{3}\sqrt{2}-67500\sqrt{3}}{2}$$

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

$$\frac{1250000\times 2\sqrt{3}-180000\sqrt{6}+5625000\sqrt{2}-810000+312500\left(\sqrt{2}\right)^{2}-45000\sqrt{2}+468750\sqrt{3}\sqrt{2}-67500\sqrt{3}}{2}$$

Multiply 1250000 and 2 to get 2500000.

$$\frac{2500000\sqrt{3}-180000\sqrt{6}+5625000\sqrt{2}-810000+312500\left(\sqrt{2}\right)^{2}-45000\sqrt{2}+468750\sqrt{3}\sqrt{2}-67500\sqrt{3}}{2}$$

The square of \sqrt{2} is 2.

$$\frac{2500000\sqrt{3}-180000\sqrt{6}+5625000\sqrt{2}-810000+312500\times 2-45000\sqrt{2}+468750\sqrt{3}\sqrt{2}-67500\sqrt{3}}{2}$$

Multiply 312500 and 2 to get 625000.

$$\frac{2500000\sqrt{3}-180000\sqrt{6}+5625000\sqrt{2}-810000+625000-45000\sqrt{2}+468750\sqrt{3}\sqrt{2}-67500\sqrt{3}}{2}$$

Add -810000 and 625000 to get -185000.

$$\frac{2500000\sqrt{3}-180000\sqrt{6}+5625000\sqrt{2}-185000-45000\sqrt{2}+468750\sqrt{3}\sqrt{2}-67500\sqrt{3}}{2}$$

Combine 5625000\sqrt{2} and -45000\sqrt{2} to get 5580000\sqrt{2}.

$$\frac{2500000\sqrt{3}-180000\sqrt{6}+5580000\sqrt{2}-185000+468750\sqrt{3}\sqrt{2}-67500\sqrt{3}}{2}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{2500000\sqrt{3}-180000\sqrt{6}+5580000\sqrt{2}-185000+468750\sqrt{6}-67500\sqrt{3}}{2}$$

Combine -180000\sqrt{6} and 468750\sqrt{6} to get 288750\sqrt{6}.

$$\frac{2500000\sqrt{3}+288750\sqrt{6}+5580000\sqrt{2}-185000-67500\sqrt{3}}{2}$$

Combine 2500000\sqrt{3} and -67500\sqrt{3} to get 2432500\sqrt{3}.

$$\frac{2432500\sqrt{3}+288750\sqrt{6}+5580000\sqrt{2}-185000}{2}$$

Factor

$$625 {(231 \sqrt{6} + 1946 \sqrt{3} + 4464 \sqrt{2} - 148)} = 6313407.715340903$$