Solve frac{2sqrt{45}+3sqrt{20}}{2sqrt{3}}*frac{sqrt{3}}{sqrt{3}}

$$\frac{ 2 \sqrt{ 45 } +3 \sqrt{ 20 } }{ 2 \sqrt{ 3 } } \times \frac{ \sqrt{ 3 } }{ \sqrt{ 3 } }$$

Evaluate

$$2\sqrt{15}\approx 7.745966692$$

Solution Steps

Divide \sqrt{3} by \sqrt{3} to get 1.

$$\frac{2\sqrt{45}+3\sqrt{20}}{2\sqrt{3}}\times 1$$

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}.

$$\frac{2\times 3\sqrt{5}+3\sqrt{20}}{2\sqrt{3}}\times 1$$

Multiply 2 and 3 to get 6.

$$\frac{6\sqrt{5}+3\sqrt{20}}{2\sqrt{3}}\times 1$$

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}.

$$\frac{6\sqrt{5}+3\times 2\sqrt{5}}{2\sqrt{3}}\times 1$$

Multiply 3 and 2 to get 6.

$$\frac{6\sqrt{5}+6\sqrt{5}}{2\sqrt{3}}\times 1$$

Combine 6\sqrt{5} and 6\sqrt{5} to get 12\sqrt{5}.

$$\frac{12\sqrt{5}}{2\sqrt{3}}\times 1$$

Cancel out 2 in both numerator and denominator.

$$\frac{6\sqrt{5}}{\sqrt{3}}\times 1$$

Rationalize the denominator of \frac{6\sqrt{5}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{6\sqrt{5}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\times 1$$

The square of \sqrt{3} is 3.

$$\frac{6\sqrt{5}\sqrt{3}}{3}\times 1$$

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

$$\frac{6\sqrt{15}}{3}\times 1$$

Divide 6\sqrt{15} by 3 to get 2\sqrt{15}.

$$2\sqrt{15}\times 1$$

Multiply 2 and 1 to get 2.

$$2\sqrt{15}$$