Solve -2sqrt{20}+42/sqrt{28}+60/sqrt{45}+2sqrt{175}

$$-2 \sqrt{ 20 } + \frac{ 42 }{ \sqrt{ 28 } } + \frac{ 60 }{ \sqrt{ 45 } } +2 \sqrt{ 175 }$$

Evaluate

$$13\sqrt{7}\approx 34.394767044$$

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}.

$$-2\times 2\sqrt{5}+\frac{42}{\sqrt{28}}+\frac{60}{\sqrt{45}}+2\sqrt{175}$$

Multiply -2 and 2 to get -4.

$$-4\sqrt{5}+\frac{42}{\sqrt{28}}+\frac{60}{\sqrt{45}}+2\sqrt{175}$$

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}.

$$-4\sqrt{5}+\frac{42}{2\sqrt{7}}+\frac{60}{\sqrt{45}}+2\sqrt{175}$$

Rationalize the denominator of \frac{42}{2\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$-4\sqrt{5}+\frac{42\sqrt{7}}{2\left(\sqrt{7}\right)^{2}}+\frac{60}{\sqrt{45}}+2\sqrt{175}$$

The square of \sqrt{7} is 7.

$$-4\sqrt{5}+\frac{42\sqrt{7}}{2\times 7}+\frac{60}{\sqrt{45}}+2\sqrt{175}$$

Cancel out 2\times 7 in both numerator and denominator.

$$-4\sqrt{5}+3\sqrt{7}+\frac{60}{\sqrt{45}}+2\sqrt{175}$$

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}.

$$-4\sqrt{5}+3\sqrt{7}+\frac{60}{3\sqrt{5}}+2\sqrt{175}$$

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

$$-4\sqrt{5}+3\sqrt{7}+\frac{60\sqrt{5}}{3\left(\sqrt{5}\right)^{2}}+2\sqrt{175}$$

The square of \sqrt{5} is 5.

$$-4\sqrt{5}+3\sqrt{7}+\frac{60\sqrt{5}}{3\times 5}+2\sqrt{175}$$

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

$$-4\sqrt{5}+3\sqrt{7}+4\sqrt{5}+2\sqrt{175}$$

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

$$3\sqrt{7}+2\sqrt{175}$$

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}.

$$3\sqrt{7}+2\times 5\sqrt{7}$$

Multiply 2 and 5 to get 10.

$$3\sqrt{7}+10\sqrt{7}$$

Combine 3\sqrt{7} and 10\sqrt{7} to get 13\sqrt{7}.

$$13\sqrt{7}$$