Solve sqrt{588}-sqrt{300}+sqrt{108}-21sqrt{3^-1}

$$\sqrt{ 588 } - \sqrt{ 300 } + \sqrt{ 108 } -21 \sqrt{ { 3 }^{ -1 } }$$

Evaluate

$$3\sqrt{3}\approx 5.196152423$$

Solution Steps

Factor 588=14^{2}\times 3. Rewrite the square root of the product \sqrt{14^{2}\times 3} as the product of square roots \sqrt{14^{2}}\sqrt{3}. Take the square root of 14^{2}.

$$14\sqrt{3}-\sqrt{300}+\sqrt{108}-21\sqrt{3^{-1}}$$

Factor 300=10^{2}\times 3. Rewrite the square root of the product \sqrt{10^{2}\times 3} as the product of square roots \sqrt{10^{2}}\sqrt{3}. Take the square root of 10^{2}.

$$14\sqrt{3}-10\sqrt{3}+\sqrt{108}-21\sqrt{3^{-1}}$$

Combine 14\sqrt{3} and -10\sqrt{3} to get 4\sqrt{3}.

$$4\sqrt{3}+\sqrt{108}-21\sqrt{3^{-1}}$$

Factor 108=6^{2}\times 3. Rewrite the square root of the product \sqrt{6^{2}\times 3} as the product of square roots \sqrt{6^{2}}\sqrt{3}. Take the square root of 6^{2}.

$$4\sqrt{3}+6\sqrt{3}-21\sqrt{3^{-1}}$$

Combine 4\sqrt{3} and 6\sqrt{3} to get 10\sqrt{3}.

$$10\sqrt{3}-21\sqrt{3^{-1}}$$

Calculate 3 to the power of -1 and get \frac{1}{3}.

$$10\sqrt{3}-21\sqrt{\frac{1}{3}}$$

Rewrite the square root of the division \sqrt{\frac{1}{3}} as the division of square roots \frac{\sqrt{1}}{\sqrt{3}}.

$$10\sqrt{3}-21\times \frac{\sqrt{1}}{\sqrt{3}}$$

Calculate the square root of 1 and get 1.

$$10\sqrt{3}-21\times \frac{1}{\sqrt{3}}$$

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

$$10\sqrt{3}-21\times \frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$10\sqrt{3}-21\times \frac{\sqrt{3}}{3}$$

Cancel out 3, the greatest common factor in 21 and 3.

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

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

$$3\sqrt{3}$$