Solve left(3sqrt{20}-sqrt{45}+2sqrt{80}+4sqrt{125}right){left(3sqrt{180}-2sqrt{245}right)}^-1

$$\left( 3 \sqrt{ 20 } - \sqrt{ 45 } +2 \sqrt{ 80 } +4 \sqrt{ 125 } \right) { \left(3 \sqrt{ 180 } -2 \sqrt{ 245 } \right) }^{ -1 }$$

Evaluate

$$\frac{31}{4}=7.75$$

Solution Steps

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}.

$$\left(3\times 2\sqrt{5}-\sqrt{45}+2\sqrt{80}+4\sqrt{125}\right)\left(3\sqrt{180}-2\sqrt{245}\right)^{-1}$$

Multiply 3 and 2 to get 6.

$$\left(6\sqrt{5}-\sqrt{45}+2\sqrt{80}+4\sqrt{125}\right)\left(3\sqrt{180}-2\sqrt{245}\right)^{-1}$$

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

$$\left(6\sqrt{5}-3\sqrt{5}+2\sqrt{80}+4\sqrt{125}\right)\left(3\sqrt{180}-2\sqrt{245}\right)^{-1}$$

Combine 6\sqrt{5} and -3\sqrt{5} to get 3\sqrt{5}.

$$\left(3\sqrt{5}+2\sqrt{80}+4\sqrt{125}\right)\left(3\sqrt{180}-2\sqrt{245}\right)^{-1}$$

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}.

$$\left(3\sqrt{5}+2\times 4\sqrt{5}+4\sqrt{125}\right)\left(3\sqrt{180}-2\sqrt{245}\right)^{-1}$$

Multiply 2 and 4 to get 8.

$$\left(3\sqrt{5}+8\sqrt{5}+4\sqrt{125}\right)\left(3\sqrt{180}-2\sqrt{245}\right)^{-1}$$

Combine 3\sqrt{5} and 8\sqrt{5} to get 11\sqrt{5}.

$$\left(11\sqrt{5}+4\sqrt{125}\right)\left(3\sqrt{180}-2\sqrt{245}\right)^{-1}$$

Factor 125=5^{2}\times 5. Rewrite the square root of the product \sqrt{5^{2}\times 5} as the product of square roots \sqrt{5^{2}}\sqrt{5}. Take the square root of 5^{2}.

$$\left(11\sqrt{5}+4\times 5\sqrt{5}\right)\left(3\sqrt{180}-2\sqrt{245}\right)^{-1}$$

Multiply 4 and 5 to get 20.

$$\left(11\sqrt{5}+20\sqrt{5}\right)\left(3\sqrt{180}-2\sqrt{245}\right)^{-1}$$

Combine 11\sqrt{5} and 20\sqrt{5} to get 31\sqrt{5}.

$$31\sqrt{5}\left(3\sqrt{180}-2\sqrt{245}\right)^{-1}$$

Factor 180=6^{2}\times 5. Rewrite the square root of the product \sqrt{6^{2}\times 5} as the product of square roots \sqrt{6^{2}}\sqrt{5}. Take the square root of 6^{2}.

$$31\sqrt{5}\left(3\times 6\sqrt{5}-2\sqrt{245}\right)^{-1}$$

Multiply 3 and 6 to get 18.

$$31\sqrt{5}\left(18\sqrt{5}-2\sqrt{245}\right)^{-1}$$

Factor 245=7^{2}\times 5. Rewrite the square root of the product \sqrt{7^{2}\times 5} as the product of square roots \sqrt{7^{2}}\sqrt{5}. Take the square root of 7^{2}.

$$31\sqrt{5}\left(18\sqrt{5}-2\times 7\sqrt{5}\right)^{-1}$$

Multiply -2 and 7 to get -14.

$$31\sqrt{5}\left(18\sqrt{5}-14\sqrt{5}\right)^{-1}$$

Combine 18\sqrt{5} and -14\sqrt{5} to get 4\sqrt{5}.

$$31\sqrt{5}\times \left(4\sqrt{5}\right)^{-1}$$

Expand \left(4\sqrt{5}\right)^{-1}.

$$31\sqrt{5}\times 4^{-1}\left(\sqrt{5}\right)^{-1}$$

Calculate 4 to the power of -1 and get \frac{1}{4}.

$$31\sqrt{5}\times \frac{1}{4}\left(\sqrt{5}\right)^{-1}$$

Multiply 31 and \frac{1}{4} to get \frac{31}{4}.

$$\frac{31}{4}\sqrt{5}\left(\sqrt{5}\right)^{-1}$$

Multiply \sqrt{5} and \left(\sqrt{5}\right)^{-1} to get 1.

$$\frac{31}{4}$$

Factor

$$\frac{31}{2 ^ {2}} = 7\frac{3}{4} = 7.75$$