Solve sqrt{33/8}-sqrt{131/2}:sqrt{3/32}

$$\sqrt { 3 \frac { 3 } { 8 } } - \sqrt { 13 \frac { 1 } { 2 } } : \sqrt { \frac { 3 } { 32 } }$$

Evaluate

$$\frac{3\sqrt{6}}{4}-12\approx -10.162882693$$

Short Solution Steps

Multiply 3 and 8 to get 24.

$$\sqrt{\frac{24+3}{8}}-\frac{\sqrt{\frac{13\times 2+1}{2}}}{\sqrt{\frac{3}{32}}}$$

Add 24 and 3 to get 27.

$$\sqrt{\frac{27}{8}}-\frac{\sqrt{\frac{13\times 2+1}{2}}}{\sqrt{\frac{3}{32}}}$$

Rewrite the square root of the division \sqrt{\frac{27}{8}} as the division of square roots \frac{\sqrt{27}}{\sqrt{8}}.

$$\frac{\sqrt{27}}{\sqrt{8}}-\frac{\sqrt{\frac{13\times 2+1}{2}}}{\sqrt{\frac{3}{32}}}$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$\frac{3\sqrt{3}}{\sqrt{8}}-\frac{\sqrt{\frac{13\times 2+1}{2}}}{\sqrt{\frac{3}{32}}}$$

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}.

$$\frac{3\sqrt{3}}{2\sqrt{2}}-\frac{\sqrt{\frac{13\times 2+1}{2}}}{\sqrt{\frac{3}{32}}}$$

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

$$\frac{3\sqrt{3}\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}-\frac{\sqrt{\frac{13\times 2+1}{2}}}{\sqrt{\frac{3}{32}}}$$

The square of \sqrt{2} is 2.

$$\frac{3\sqrt{3}\sqrt{2}}{2\times 2}-\frac{\sqrt{\frac{13\times 2+1}{2}}}{\sqrt{\frac{3}{32}}}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{3\sqrt{6}}{2\times 2}-\frac{\sqrt{\frac{13\times 2+1}{2}}}{\sqrt{\frac{3}{32}}}$$

Multiply 2 and 2 to get 4.

$$\frac{3\sqrt{6}}{4}-\frac{\sqrt{\frac{13\times 2+1}{2}}}{\sqrt{\frac{3}{32}}}$$

Multiply 13 and 2 to get 26.

$$\frac{3\sqrt{6}}{4}-\frac{\sqrt{\frac{26+1}{2}}}{\sqrt{\frac{3}{32}}}$$

Add 26 and 1 to get 27.

$$\frac{3\sqrt{6}}{4}-\frac{\sqrt{\frac{27}{2}}}{\sqrt{\frac{3}{32}}}$$

Rewrite the square root of the division \sqrt{\frac{27}{2}} as the division of square roots \frac{\sqrt{27}}{\sqrt{2}}.

$$\frac{3\sqrt{6}}{4}-\frac{\frac{\sqrt{27}}{\sqrt{2}}}{\sqrt{\frac{3}{32}}}$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$\frac{3\sqrt{6}}{4}-\frac{\frac{3\sqrt{3}}{\sqrt{2}}}{\sqrt{\frac{3}{32}}}$$

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

$$\frac{3\sqrt{6}}{4}-\frac{\frac{3\sqrt{3}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}}{\sqrt{\frac{3}{32}}}$$

The square of \sqrt{2} is 2.

$$\frac{3\sqrt{6}}{4}-\frac{\frac{3\sqrt{3}\sqrt{2}}{2}}{\sqrt{\frac{3}{32}}}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{3\sqrt{6}}{4}-\frac{\frac{3\sqrt{6}}{2}}{\sqrt{\frac{3}{32}}}$$

Rewrite the square root of the division \sqrt{\frac{3}{32}} as the division of square roots \frac{\sqrt{3}}{\sqrt{32}}.

$$\frac{3\sqrt{6}}{4}-\frac{\frac{3\sqrt{6}}{2}}{\frac{\sqrt{3}}{\sqrt{32}}}$$

Factor 32=4^{2}\times 2. Rewrite the square root of the product \sqrt{4^{2}\times 2} as the product of square roots \sqrt{4^{2}}\sqrt{2}. Take the square root of 4^{2}.

$$\frac{3\sqrt{6}}{4}-\frac{\frac{3\sqrt{6}}{2}}{\frac{\sqrt{3}}{4\sqrt{2}}}$$

Rationalize the denominator of \frac{\sqrt{3}}{4\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{3\sqrt{6}}{4}-\frac{\frac{3\sqrt{6}}{2}}{\frac{\sqrt{3}\sqrt{2}}{4\left(\sqrt{2}\right)^{2}}}$$

The square of \sqrt{2} is 2.

$$\frac{3\sqrt{6}}{4}-\frac{\frac{3\sqrt{6}}{2}}{\frac{\sqrt{3}\sqrt{2}}{4\times 2}}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{3\sqrt{6}}{4}-\frac{\frac{3\sqrt{6}}{2}}{\frac{\sqrt{6}}{4\times 2}}$$

Multiply 4 and 2 to get 8.

$$\frac{3\sqrt{6}}{4}-\frac{\frac{3\sqrt{6}}{2}}{\frac{\sqrt{6}}{8}}$$

Divide \frac{3\sqrt{6}}{2} by \frac{\sqrt{6}}{8} by multiplying \frac{3\sqrt{6}}{2} by the reciprocal of \frac{\sqrt{6}}{8}.

$$\frac{3\sqrt{6}}{4}-\frac{3\sqrt{6}\times 8}{2\sqrt{6}}$$

Cancel out 2\sqrt{6} in both numerator and denominator.

$$\frac{3\sqrt{6}}{4}-3\times 4$$

Multiply 3 and 4 to get 12.

$$\frac{3\sqrt{6}}{4}-12$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 12 times \frac{4}{4}.

$$\frac{3\sqrt{6}}{4}-\frac{12\times 4}{4}$$

Since \frac{3\sqrt{6}}{4} and \frac{12\times 4}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\sqrt{6}-12\times 4}{4}$$

Do the multiplications in 3\sqrt{6}-12\times 4.

$$\frac{3\sqrt{6}-48}{4}$$