Solve (3/4-1)^-3-3sqrt{3/8+3}*(-1/6)^2=

$$( \frac { 3 } { 4 } - 1 ) ^ { - 3 - 3 } \sqrt { \frac { 3 } { 8 } + 3 } \cdot ( - \frac { 1 } { 6 } ) ^ { 2 } =$$

Evaluate

$$\frac{256\sqrt{6}}{3}\approx 209.023124717$$

Solution Steps

Subtract 1 from \frac{3}{4} to get -\frac{1}{4}.

$$\left(-\frac{1}{4}\right)^{-3-3}\sqrt{\frac{3}{8}+3}\left(-\frac{1}{6}\right)^{2}$$

Subtract 3 from -3 to get -6.

$$\left(-\frac{1}{4}\right)^{-6}\sqrt{\frac{3}{8}+3}\left(-\frac{1}{6}\right)^{2}$$

Calculate -\frac{1}{4} to the power of -6 and get 4096.

$$4096\sqrt{\frac{3}{8}+3}\left(-\frac{1}{6}\right)^{2}$$

Add \frac{3}{8} and 3 to get \frac{27}{8}.

$$4096\sqrt{\frac{27}{8}}\left(-\frac{1}{6}\right)^{2}$$

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

$$4096\times \frac{\sqrt{27}}{\sqrt{8}}\left(-\frac{1}{6}\right)^{2}$$

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}.

$$4096\times \frac{3\sqrt{3}}{\sqrt{8}}\left(-\frac{1}{6}\right)^{2}$$

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}.

$$4096\times \frac{3\sqrt{3}}{2\sqrt{2}}\left(-\frac{1}{6}\right)^{2}$$

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

$$4096\times \frac{3\sqrt{3}\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}\left(-\frac{1}{6}\right)^{2}$$

The square of \sqrt{2} is 2.

$$4096\times \frac{3\sqrt{3}\sqrt{2}}{2\times 2}\left(-\frac{1}{6}\right)^{2}$$

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

$$4096\times \frac{3\sqrt{6}}{2\times 2}\left(-\frac{1}{6}\right)^{2}$$

Multiply 2 and 2 to get 4.

$$4096\times \frac{3\sqrt{6}}{4}\left(-\frac{1}{6}\right)^{2}$$

Calculate -\frac{1}{6} to the power of 2 and get \frac{1}{36}.

$$4096\times \frac{3\sqrt{6}}{4}\times \frac{1}{36}$$

Multiply 4096 and \frac{1}{36} to get \frac{1024}{9}.

$$\frac{1024}{9}\times \frac{3\sqrt{6}}{4}$$

Multiply \frac{1024}{9} times \frac{3\sqrt{6}}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1024\times 3\sqrt{6}}{9\times 4}$$

Cancel out 3\times 4 in both numerator and denominator.

$$\frac{256\sqrt{6}}{3}$$