Solve sqrt{(15/4)^2+(frac{3sqrt{7}}{4})^2}

$$\sqrt { ( \frac { 15 } { 4 } ) ^ { 2 } + ( \frac { 3 \sqrt { 7 } } { 4 } ) ^ { 2 } }$$

Evaluate

$$3\sqrt{2}\approx 4.242640687$$

Solution Steps

Calculate \frac{15}{4} to the power of 2 and get \frac{225}{16}.

$$\sqrt{\frac{225}{16}+\left(\frac{3\sqrt{7}}{4}\right)^{2}}$$

To raise \frac{3\sqrt{7}}{4} to a power, raise both numerator and denominator to the power and then divide.

$$\sqrt{\frac{225}{16}+\frac{\left(3\sqrt{7}\right)^{2}}{4^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Expand 4^{2}.

$$\sqrt{\frac{225}{16}+\frac{\left(3\sqrt{7}\right)^{2}}{16}}$$

Since \frac{225}{16} and \frac{\left(3\sqrt{7}\right)^{2}}{16} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{225+\left(3\sqrt{7}\right)^{2}}{16}}$$

Expand \left(3\sqrt{7}\right)^{2}.

$$\sqrt{\frac{225+3^{2}\left(\sqrt{7}\right)^{2}}{16}}$$

Calculate 3 to the power of 2 and get 9.

$$\sqrt{\frac{225+9\left(\sqrt{7}\right)^{2}}{16}}$$

The square of \sqrt{7} is 7.

$$\sqrt{\frac{225+9\times 7}{16}}$$

Multiply 9 and 7 to get 63.

$$\sqrt{\frac{225+63}{16}}$$

Add 225 and 63 to get 288.

$$\sqrt{\frac{288}{16}}$$

Divide 288 by 16 to get 18.

$$\sqrt{18}$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$3\sqrt{2}$$