Solve 45/sqrt{28}+60/sqrt{45}-2sqrt{20}+2sqrt{175}

$$\frac { 45 } { \sqrt { 28 } } + \frac { 60 } { \sqrt { 45 } } - 2 \sqrt { 20 } + 2 \sqrt { 175 }$$

Evaluate

$$\frac{185\sqrt{7}}{14}\approx 34.961713753$$

Solution Steps

Factor 28=2^{2}\times 7. Rewrite the square root of the product \sqrt{2^{2}\times 7} as the product of square roots \sqrt{2^{2}}\sqrt{7}. Take the square root of 2^{2}.

$$\frac{45}{2\sqrt{7}}+\frac{60}{\sqrt{45}}-2\sqrt{20}+2\sqrt{175}$$

Rationalize the denominator of \frac{45}{2\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\frac{45\sqrt{7}}{2\left(\sqrt{7}\right)^{2}}+\frac{60}{\sqrt{45}}-2\sqrt{20}+2\sqrt{175}$$

The square of \sqrt{7} is 7.

$$\frac{45\sqrt{7}}{2\times 7}+\frac{60}{\sqrt{45}}-2\sqrt{20}+2\sqrt{175}$$

Multiply 2 and 7 to get 14.

$$\frac{45\sqrt{7}}{14}+\frac{60}{\sqrt{45}}-2\sqrt{20}+2\sqrt{175}$$

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}.

$$\frac{45\sqrt{7}}{14}+\frac{60}{3\sqrt{5}}-2\sqrt{20}+2\sqrt{175}$$

Rationalize the denominator of \frac{60}{3\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{45\sqrt{7}}{14}+\frac{60\sqrt{5}}{3\left(\sqrt{5}\right)^{2}}-2\sqrt{20}+2\sqrt{175}$$

The square of \sqrt{5} is 5.

$$\frac{45\sqrt{7}}{14}+\frac{60\sqrt{5}}{3\times 5}-2\sqrt{20}+2\sqrt{175}$$

Cancel out 3\times 5 in both numerator and denominator.

$$\frac{45\sqrt{7}}{14}+4\sqrt{5}-2\sqrt{20}+2\sqrt{175}$$

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}.

$$\frac{45\sqrt{7}}{14}+4\sqrt{5}-2\times 2\sqrt{5}+2\sqrt{175}$$

Multiply -2 and 2 to get -4.

$$\frac{45\sqrt{7}}{14}+4\sqrt{5}-4\sqrt{5}+2\sqrt{175}$$

Combine 4\sqrt{5} and -4\sqrt{5} to get 0.

$$\frac{45\sqrt{7}}{14}+2\sqrt{175}$$

Factor 175=5^{2}\times 7. Rewrite the square root of the product \sqrt{5^{2}\times 7} as the product of square roots \sqrt{5^{2}}\sqrt{7}. Take the square root of 5^{2}.

$$\frac{45\sqrt{7}}{14}+2\times 5\sqrt{7}$$

Multiply 2 and 5 to get 10.

$$\frac{45\sqrt{7}}{14}+10\sqrt{7}$$

Combine \frac{45\sqrt{7}}{14} and 10\sqrt{7} to get \frac{185}{14}\sqrt{7}.

$$\frac{185}{14}\sqrt{7}$$