Solve frac{sqrt{125}-sqrt{20}+5sqrt{45}}{sqrt{3}}

$$\frac { \sqrt { 125 } - \sqrt { 20 } + 5 \sqrt { 45 } } { \sqrt { 3 } }$$

Evaluate

$$6\sqrt{15}\approx 23.237900077$$

Solution Steps

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}.

$$\frac{5\sqrt{5}-\sqrt{20}+5\sqrt{45}}{\sqrt{3}}$$

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}.

$$\frac{5\sqrt{5}-2\sqrt{5}+5\sqrt{45}}{\sqrt{3}}$$

Combine 5\sqrt{5} and -2\sqrt{5} to get 3\sqrt{5}.

$$\frac{3\sqrt{5}+5\sqrt{45}}{\sqrt{3}}$$

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}.

$$\frac{3\sqrt{5}+5\times 3\sqrt{5}}{\sqrt{3}}$$

Multiply 5 and 3 to get 15.

$$\frac{3\sqrt{5}+15\sqrt{5}}{\sqrt{3}}$$

Combine 3\sqrt{5} and 15\sqrt{5} to get 18\sqrt{5}.

$$\frac{18\sqrt{5}}{\sqrt{3}}$$

Rationalize the denominator of \frac{18\sqrt{5}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{18\sqrt{5}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{18\sqrt{5}\sqrt{3}}{3}$$

To multiply \sqrt{5} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{18\sqrt{15}}{3}$$

Divide 18\sqrt{15} by 3 to get 6\sqrt{15}.

$$6\sqrt{15}$$