Solve frac{sqrt{2}+frac{sqrt{45}}{sqrt{3*5}}}{sqrt{3}}=

$$\frac { \sqrt { 2 } + \frac { \sqrt { 45 } } { \sqrt { 3 \cdot 5 } } } { \sqrt { 3 } } =$$

Evaluate

$$\frac{\sqrt{6}}{3}+1\approx 1.816496581$$

Solution Steps

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}.

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

Multiply 3 and 5 to get 15.

$$\frac{\sqrt{2}+\frac{3\sqrt{5}}{\sqrt{15}}}{\sqrt{3}}$$

Rationalize the denominator of \frac{3\sqrt{5}}{\sqrt{15}} by multiplying numerator and denominator by \sqrt{15}.

$$\frac{\sqrt{2}+\frac{3\sqrt{5}\sqrt{15}}{\left(\sqrt{15}\right)^{2}}}{\sqrt{3}}$$

The square of \sqrt{15} is 15.

$$\frac{\sqrt{2}+\frac{3\sqrt{5}\sqrt{15}}{15}}{\sqrt{3}}$$

Factor 15=5\times 3. Rewrite the square root of the product \sqrt{5\times 3} as the product of square roots \sqrt{5}\sqrt{3}.

$$\frac{\sqrt{2}+\frac{3\sqrt{5}\sqrt{5}\sqrt{3}}{15}}{\sqrt{3}}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$\frac{\sqrt{2}+\frac{3\times 5\sqrt{3}}{15}}{\sqrt{3}}$$

Multiply 3 and 5 to get 15.

$$\frac{\sqrt{2}+\frac{15\sqrt{3}}{15}}{\sqrt{3}}$$

Cancel out 15 and 15.

$$\frac{\sqrt{2}+\sqrt{3}}{\sqrt{3}}$$

Rationalize the denominator of \frac{\sqrt{2}+\sqrt{3}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

$$\frac{\left(\sqrt{2}+\sqrt{3}\right)\sqrt{3}}{3}$$

Use the distributive property to multiply \sqrt{2}+\sqrt{3} by \sqrt{3}.

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

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\sqrt{6}+\left(\sqrt{3}\right)^{2}}{3}$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{6}+3}{3}$$