Solve sqrt{3}-sqrt{28}+21sqrt{1/63}

$$\sqrt { 3 } - \sqrt { 28 } + 21 \sqrt { \frac { 1 } { 63 } }$$

Evaluate

$$\sqrt{3}-\sqrt{7}\approx -0.913700503$$

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}.

$$\sqrt{3}-2\sqrt{7}+21\sqrt{\frac{1}{63}}$$

Rewrite the square root of the division \sqrt{\frac{1}{63}} as the division of square roots \frac{\sqrt{1}}{\sqrt{63}}.

$$\sqrt{3}-2\sqrt{7}+21\times \frac{\sqrt{1}}{\sqrt{63}}$$

Calculate the square root of 1 and get 1.

$$\sqrt{3}-2\sqrt{7}+21\times \frac{1}{\sqrt{63}}$$

Factor 63=3^{2}\times 7. Rewrite the square root of the product \sqrt{3^{2}\times 7} as the product of square roots \sqrt{3^{2}}\sqrt{7}. Take the square root of 3^{2}.

$$\sqrt{3}-2\sqrt{7}+21\times \frac{1}{3\sqrt{7}}$$

Rationalize the denominator of \frac{1}{3\sqrt{7}} by multiplying numerator and denominator by \sqrt{7}.

$$\sqrt{3}-2\sqrt{7}+21\times \frac{\sqrt{7}}{3\left(\sqrt{7}\right)^{2}}$$

The square of \sqrt{7} is 7.

$$\sqrt{3}-2\sqrt{7}+21\times \frac{\sqrt{7}}{3\times 7}$$

Multiply 3 and 7 to get 21.

$$\sqrt{3}-2\sqrt{7}+21\times \frac{\sqrt{7}}{21}$$

Cancel out 21 and 21.

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

Combine -2\sqrt{7} and \sqrt{7} to get -\sqrt{7}.

$$\sqrt{3}-\sqrt{7}$$