Solve sqrt{27}+sqrt{125}-sqrt{2187}+sqrt{3125}

$$\sqrt { 27 } + \sqrt { 125 } - \sqrt { 2187 } + \sqrt { 3125 }$$

Evaluate

$$6\left(5\sqrt{5}-4\sqrt{3}\right)\approx 25.512819943$$

Solution Steps

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$3\sqrt{3}+\sqrt{125}-\sqrt{2187}+\sqrt{3125}$$

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}.

$$3\sqrt{3}+5\sqrt{5}-\sqrt{2187}+\sqrt{3125}$$

Factor 2187=27^{2}\times 3. Rewrite the square root of the product \sqrt{27^{2}\times 3} as the product of square roots \sqrt{27^{2}}\sqrt{3}. Take the square root of 27^{2}.

$$3\sqrt{3}+5\sqrt{5}-27\sqrt{3}+\sqrt{3125}$$

Combine 3\sqrt{3} and -27\sqrt{3} to get -24\sqrt{3}.

$$-24\sqrt{3}+5\sqrt{5}+\sqrt{3125}$$

Factor 3125=25^{2}\times 5. Rewrite the square root of the product \sqrt{25^{2}\times 5} as the product of square roots \sqrt{25^{2}}\sqrt{5}. Take the square root of 25^{2}.

$$-24\sqrt{3}+5\sqrt{5}+25\sqrt{5}$$

Combine 5\sqrt{5} and 25\sqrt{5} to get 30\sqrt{5}.

$$-24\sqrt{3}+30\sqrt{5}$$