Solve sqrt{80}-2sqrt{252}+3sqrt{405}-3sqrt{500}

$$\sqrt { 80 } - 2 \sqrt { 252 } + 3 \sqrt { 405 } - 3 \sqrt { 500 }$$

Evaluate

$$\sqrt{5}-12\sqrt{7}\approx -29.512947755$$

Solution Steps

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}.

$$4\sqrt{5}-2\sqrt{252}+3\sqrt{405}-3\sqrt{500}$$

Factor 252=6^{2}\times 7. Rewrite the square root of the product \sqrt{6^{2}\times 7} as the product of square roots \sqrt{6^{2}}\sqrt{7}. Take the square root of 6^{2}.

$$4\sqrt{5}-2\times 6\sqrt{7}+3\sqrt{405}-3\sqrt{500}$$

Multiply -2 and 6 to get -12.

$$4\sqrt{5}-12\sqrt{7}+3\sqrt{405}-3\sqrt{500}$$

Factor 405=9^{2}\times 5. Rewrite the square root of the product \sqrt{9^{2}\times 5} as the product of square roots \sqrt{9^{2}}\sqrt{5}. Take the square root of 9^{2}.

$$4\sqrt{5}-12\sqrt{7}+3\times 9\sqrt{5}-3\sqrt{500}$$

Multiply 3 and 9 to get 27.

$$4\sqrt{5}-12\sqrt{7}+27\sqrt{5}-3\sqrt{500}$$

Combine 4\sqrt{5} and 27\sqrt{5} to get 31\sqrt{5}.

$$31\sqrt{5}-12\sqrt{7}-3\sqrt{500}$$

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}.

$$31\sqrt{5}-12\sqrt{7}-3\times 10\sqrt{5}$$

Multiply -3 and 10 to get -30.

$$31\sqrt{5}-12\sqrt{7}-30\sqrt{5}$$

Combine 31\sqrt{5} and -30\sqrt{5} to get \sqrt{5}.

$$\sqrt{5}-12\sqrt{7}$$