Solve sqrt{450}-4sqrt{320}+3sqrt{80}-5sqrt{500}

$$\sqrt { 450 } - 4 \sqrt { 320 } + 3 \sqrt { 80 } - 5 \sqrt { 500 }$$

Evaluate

$$15\sqrt{2}-70\sqrt{5}\approx -135.311554989$$

Solution Steps

Factor 450=15^{2}\times 2. Rewrite the square root of the product \sqrt{15^{2}\times 2} as the product of square roots \sqrt{15^{2}}\sqrt{2}. Take the square root of 15^{2}.

$$15\sqrt{2}-4\sqrt{320}+3\sqrt{80}-5\sqrt{500}$$

Factor 320=8^{2}\times 5. Rewrite the square root of the product \sqrt{8^{2}\times 5} as the product of square roots \sqrt{8^{2}}\sqrt{5}. Take the square root of 8^{2}.

$$15\sqrt{2}-4\times 8\sqrt{5}+3\sqrt{80}-5\sqrt{500}$$

Multiply -4 and 8 to get -32.

$$15\sqrt{2}-32\sqrt{5}+3\sqrt{80}-5\sqrt{500}$$

Factor 80=4^{2}\times 5. Rewrite the square root of the product \sqrt{4^{2}\times 5} as the product of square roots \sqrt{4^{2}}\sqrt{5}. Take the square root of 4^{2}.

$$15\sqrt{2}-32\sqrt{5}+3\times 4\sqrt{5}-5\sqrt{500}$$

Multiply 3 and 4 to get 12.

$$15\sqrt{2}-32\sqrt{5}+12\sqrt{5}-5\sqrt{500}$$

Combine -32\sqrt{5} and 12\sqrt{5} to get -20\sqrt{5}.

$$15\sqrt{2}-20\sqrt{5}-5\sqrt{500}$$

Factor 500=10^{2}\times 5. Rewrite the square root of the product \sqrt{10^{2}\times 5} as the product of square roots \sqrt{10^{2}}\sqrt{5}. Take the square root of 10^{2}.

$$15\sqrt{2}-20\sqrt{5}-5\times 10\sqrt{5}$$

Multiply -5 and 10 to get -50.

$$15\sqrt{2}-20\sqrt{5}-50\sqrt{5}$$

Combine -20\sqrt{5} and -50\sqrt{5} to get -70\sqrt{5}.

$$15\sqrt{2}-70\sqrt{5}$$