Solve sqrt{3^2+(3/2sqrt{3})^2}

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

Evaluate

$$\frac{3\sqrt{7}}{2}\approx 3.968626967$$

Solution Steps

Calculate 3 to the power of 2 and get 9.

$$\sqrt{9+\left(\frac{3}{2}\sqrt{3}\right)^{2}}$$

Expand \left(\frac{3}{2}\sqrt{3}\right)^{2}.

$$\sqrt{9+\left(\frac{3}{2}\right)^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate \frac{3}{2} to the power of 2 and get \frac{9}{4}.

$$\sqrt{9+\frac{9}{4}\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\sqrt{9+\frac{9}{4}\times 3}$$

Multiply \frac{9}{4} and 3 to get \frac{27}{4}.

$$\sqrt{9+\frac{27}{4}}$$

Add 9 and \frac{27}{4} to get \frac{63}{4}.

$$\sqrt{\frac{63}{4}}$$

Rewrite the square root of the division \sqrt{\frac{63}{4}} as the division of square roots \frac{\sqrt{63}}{\sqrt{4}}.

$$\frac{\sqrt{63}}{\sqrt{4}}$$

Factor 63=3^{2}\times 7. Rewrite the square root of the product \sqrt{3^{2}\times 7} as the product of square roots \sqrt{3^{2}}\sqrt{7}. Take the square root of 3^{2}.

$$\frac{3\sqrt{7}}{\sqrt{4}}$$

Calculate the square root of 4 and get 2.

$$\frac{3\sqrt{7}}{2}$$