Solve sqrt{(5/2)^2+25/3}

$$\sqrt { ( \frac { 5 } { 2 } ) ^ { 2 } + \frac { 25 } { 3 } }$$

Evaluate

$$\frac{5\sqrt{21}}{6}\approx 3.818813079$$

Solution Steps

Calculate \frac{5}{2} to the power of 2 and get \frac{25}{4}.

$$\sqrt{\frac{25}{4}+\frac{25}{3}}$$

Least common multiple of 4 and 3 is 12. Convert \frac{25}{4} and \frac{25}{3} to fractions with denominator 12.

$$\sqrt{\frac{75}{12}+\frac{100}{12}}$$

Since \frac{75}{12} and \frac{100}{12} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{75+100}{12}}$$

Add 75 and 100 to get 175.

$$\sqrt{\frac{175}{12}}$$

Rewrite the square root of the division \sqrt{\frac{175}{12}} as the division of square roots \frac{\sqrt{175}}{\sqrt{12}}.

$$\frac{\sqrt{175}}{\sqrt{12}}$$

Factor 175=5^{2}\times 7. Rewrite the square root of the product \sqrt{5^{2}\times 7} as the product of square roots \sqrt{5^{2}}\sqrt{7}. Take the square root of 5^{2}.

$$\frac{5\sqrt{7}}{\sqrt{12}}$$

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$\frac{5\sqrt{7}}{2\sqrt{3}}$$

Rationalize the denominator of \frac{5\sqrt{7}}{2\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

$$\frac{5\sqrt{7}\sqrt{3}}{2\times 3}$$

To multiply \sqrt{7} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{5\sqrt{21}}{2\times 3}$$

Multiply 2 and 3 to get 6.

$$\frac{5\sqrt{21}}{6}$$