Solve sqrt{7/4}+sqrt{9/16}+sqrt{25/76}=

$$\sqrt { \frac { 7 } { 4 } } + \sqrt { \frac { 9 } { 16 } } + \sqrt { \frac { 25 } { 76 } } =$$

Evaluate

$$\frac{\sqrt{7}}{2}+\frac{5\sqrt{19}}{38}+\frac{3}{4}\approx 2.64641499$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{7}{4}} as the division of square roots \frac{\sqrt{7}}{\sqrt{4}}.

$$\frac{\sqrt{7}}{\sqrt{4}}+\sqrt{\frac{9}{16}}+\sqrt{\frac{25}{76}}$$

Calculate the square root of 4 and get 2.

$$\frac{\sqrt{7}}{2}+\sqrt{\frac{9}{16}}+\sqrt{\frac{25}{76}}$$

Rewrite the square root of the division \frac{9}{16} as the division of square roots \frac{\sqrt{9}}{\sqrt{16}}. Take the square root of both numerator and denominator.

$$\frac{\sqrt{7}}{2}+\frac{3}{4}+\sqrt{\frac{25}{76}}$$

Rewrite the square root of the division \sqrt{\frac{25}{76}} as the division of square roots \frac{\sqrt{25}}{\sqrt{76}}.

$$\frac{\sqrt{7}}{2}+\frac{3}{4}+\frac{\sqrt{25}}{\sqrt{76}}$$

Calculate the square root of 25 and get 5.

$$\frac{\sqrt{7}}{2}+\frac{3}{4}+\frac{5}{\sqrt{76}}$$

Factor 76=2^{2}\times 19. Rewrite the square root of the product \sqrt{2^{2}\times 19} as the product of square roots \sqrt{2^{2}}\sqrt{19}. Take the square root of 2^{2}.

$$\frac{\sqrt{7}}{2}+\frac{3}{4}+\frac{5}{2\sqrt{19}}$$

Rationalize the denominator of \frac{5}{2\sqrt{19}} by multiplying numerator and denominator by \sqrt{19}.

$$\frac{\sqrt{7}}{2}+\frac{3}{4}+\frac{5\sqrt{19}}{2\left(\sqrt{19}\right)^{2}}$$

The square of \sqrt{19} is 19.

$$\frac{\sqrt{7}}{2}+\frac{3}{4}+\frac{5\sqrt{19}}{2\times 19}$$

Multiply 2 and 19 to get 38.

$$\frac{\sqrt{7}}{2}+\frac{3}{4}+\frac{5\sqrt{19}}{38}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 4 is 4. Multiply \frac{\sqrt{7}}{2} times \frac{2}{2}.

$$\frac{2\sqrt{7}}{4}+\frac{3}{4}+\frac{5\sqrt{19}}{38}$$

Since \frac{2\sqrt{7}}{4} and \frac{3}{4} have the same denominator, add them by adding their numerators.

$$\frac{2\sqrt{7}+3}{4}+\frac{5\sqrt{19}}{38}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4 and 38 is 76. Multiply \frac{2\sqrt{7}+3}{4} times \frac{19}{19}. Multiply \frac{5\sqrt{19}}{38} times \frac{2}{2}.

$$\frac{19\left(2\sqrt{7}+3\right)}{76}+\frac{2\times 5\sqrt{19}}{76}$$

Since \frac{19\left(2\sqrt{7}+3\right)}{76} and \frac{2\times 5\sqrt{19}}{76} have the same denominator, add them by adding their numerators.

$$\frac{19\left(2\sqrt{7}+3\right)+2\times 5\sqrt{19}}{76}$$

Do the multiplications in 19\left(2\sqrt{7}+3\right)+2\times 5\sqrt{19}.

$$\frac{38\sqrt{7}+57+10\sqrt{19}}{76}$$