Solve 1/6*sqrt{32}-frac{6*sqrt{32}-4}{6*sqrt{32}*1/8}

$$\frac{ 1 }{ 6 \times \sqrt{ 32 } } - \frac{ 6 \times \sqrt{ 32 } -4 }{ 6 \times \sqrt{ 32 } \times \frac{ 1 }{ 8 } }$$

Evaluate

$$\frac{11\sqrt{2}}{16}-8\approx -7.027728176$$

Short Solution Steps

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}.

$$\frac{1}{6\times 4\sqrt{2}}-\frac{6\sqrt{32}-4}{6\sqrt{32}\times \frac{1}{8}}$$

Multiply 6 and 4 to get 24.

$$\frac{1}{24\sqrt{2}}-\frac{6\sqrt{32}-4}{6\sqrt{32}\times \frac{1}{8}}$$

Rationalize the denominator of \frac{1}{24\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\sqrt{2}}{24\left(\sqrt{2}\right)^{2}}-\frac{6\sqrt{32}-4}{6\sqrt{32}\times \frac{1}{8}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{2}}{24\times 2}-\frac{6\sqrt{32}-4}{6\sqrt{32}\times \frac{1}{8}}$$

Multiply 24 and 2 to get 48.

$$\frac{\sqrt{2}}{48}-\frac{6\sqrt{32}-4}{6\sqrt{32}\times \frac{1}{8}}$$

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}.

$$\frac{\sqrt{2}}{48}-\frac{6\times 4\sqrt{2}-4}{6\sqrt{32}\times \frac{1}{8}}$$

Multiply 6 and 4 to get 24.

$$\frac{\sqrt{2}}{48}-\frac{24\sqrt{2}-4}{6\sqrt{32}\times \frac{1}{8}}$$

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}.

$$\frac{\sqrt{2}}{48}-\frac{24\sqrt{2}-4}{6\times 4\sqrt{2}\times \frac{1}{8}}$$

Multiply 6 and 4 to get 24.

$$\frac{\sqrt{2}}{48}-\frac{24\sqrt{2}-4}{24\sqrt{2}\times \frac{1}{8}}$$

Multiply 24 and \frac{1}{8} to get \frac{24}{8}.

$$\frac{\sqrt{2}}{48}-\frac{24\sqrt{2}-4}{\frac{24}{8}\sqrt{2}}$$

Divide 24 by 8 to get 3.

$$\frac{\sqrt{2}}{48}-\frac{24\sqrt{2}-4}{3\sqrt{2}}$$

Rationalize the denominator of \frac{24\sqrt{2}-4}{3\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\sqrt{2}}{48}-\frac{\left(24\sqrt{2}-4\right)\sqrt{2}}{3\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{2}}{48}-\frac{\left(24\sqrt{2}-4\right)\sqrt{2}}{3\times 2}$$

Multiply 3 and 2 to get 6.

$$\frac{\sqrt{2}}{48}-\frac{\left(24\sqrt{2}-4\right)\sqrt{2}}{6}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 48 and 6 is 48. Multiply \frac{\left(24\sqrt{2}-4\right)\sqrt{2}}{6} times \frac{8}{8}.

$$\frac{\sqrt{2}}{48}-\frac{8\left(24\sqrt{2}-4\right)\sqrt{2}}{48}$$

Since \frac{\sqrt{2}}{48} and \frac{8\left(24\sqrt{2}-4\right)\sqrt{2}}{48} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\sqrt{2}-8\left(24\sqrt{2}-4\right)\sqrt{2}}{48}$$

Do the multiplications in \sqrt{2}-8\left(24\sqrt{2}-4\right)\sqrt{2}.

$$\frac{\sqrt{2}-384+32\sqrt{2}}{48}$$

Do the calculations in \sqrt{2}-384+32\sqrt{2}.

$$\frac{33\sqrt{2}-384}{48}$$

Factor

$$\frac{11 \sqrt{2} - 128}{16} = -7.027728175868497$$