Solve sqrt{21/4}-sqrt{155/8}

$$\sqrt { 2 \frac { 1 } { 4 } } - \sqrt { 15 \frac { 5 } { 8 } }$$

Evaluate

$$-\frac{5\sqrt{10}}{4}+\frac{3}{2}\approx -2.452847075$$

Short Solution Steps

Multiply 2 and 4 to get 8.

$$\sqrt{\frac{8+1}{4}}-\sqrt{\frac{15\times 8+5}{8}}$$

Add 8 and 1 to get 9.

$$\sqrt{\frac{9}{4}}-\sqrt{\frac{15\times 8+5}{8}}$$

Rewrite the square root of the division \frac{9}{4} as the division of square roots \frac{\sqrt{9}}{\sqrt{4}}. Take the square root of both numerator and denominator.

$$\frac{3}{2}-\sqrt{\frac{15\times 8+5}{8}}$$

Multiply 15 and 8 to get 120.

$$\frac{3}{2}-\sqrt{\frac{120+5}{8}}$$

Add 120 and 5 to get 125.

$$\frac{3}{2}-\sqrt{\frac{125}{8}}$$

Rewrite the square root of the division \sqrt{\frac{125}{8}} as the division of square roots \frac{\sqrt{125}}{\sqrt{8}}.

$$\frac{3}{2}-\frac{\sqrt{125}}{\sqrt{8}}$$

Factor 125=5^{2}\times 5. Rewrite the square root of the product \sqrt{5^{2}\times 5} as the product of square roots \sqrt{5^{2}}\sqrt{5}. Take the square root of 5^{2}.

$$\frac{3}{2}-\frac{5\sqrt{5}}{\sqrt{8}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

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

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

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

The square of \sqrt{2} is 2.

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

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{3}{2}-\frac{5\sqrt{10}}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$\frac{3}{2}-\frac{5\sqrt{10}}{4}$$

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

$$\frac{3\times 2}{4}-\frac{5\sqrt{10}}{4}$$

Since \frac{3\times 2}{4} and \frac{5\sqrt{10}}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\times 2-5\sqrt{10}}{4}$$

Do the multiplications in 3\times 2-5\sqrt{10}.

$$\frac{6-5\sqrt{10}}{4}$$