Solve frac{{left(sqrt{3}right)}^-2}{{left(1/sqrt{3}right)}^-2}-{left(log_{2}({32})right)}^-1+2/7*{left(1/7right)}^-1

$$\frac{ { \left( \sqrt{ 3 } \right) }^{ -2 } }{ { \left( \frac{ 1 }{ \sqrt{ 3 } } \right) }^{ -2 } } - { \left( \log_{ 2 }({ 32 }) \right) }^{ -1 } + \frac{ 2 }{ 7 } \times { \left( \frac{ 1 }{ 7 } \right) }^{ -1 }$$

Evaluate

$$\frac{86}{45}\approx 1.911111111$$

Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\left(\sqrt{3}\right)^{-2}}{\left(\frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\right)^{-2}}-\left(\log_{2}\left(32\right)\right)^{-1}+\frac{2}{7}\times \left(\frac{1}{7}\right)^{-1}$$

The square of \sqrt{3} is 3.

$$\frac{\left(\sqrt{3}\right)^{-2}}{\left(\frac{\sqrt{3}}{3}\right)^{-2}}-\left(\log_{2}\left(32\right)\right)^{-1}+\frac{2}{7}\times \left(\frac{1}{7}\right)^{-1}$$

To raise \frac{\sqrt{3}}{3} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(\sqrt{3}\right)^{-2}}{\frac{\left(\sqrt{3}\right)^{-2}}{3^{-2}}}-\left(\log_{2}\left(32\right)\right)^{-1}+\frac{2}{7}\times \left(\frac{1}{7}\right)^{-1}$$

Calculate 3 to the power of -2 and get \frac{1}{9}.

$$\frac{\left(\sqrt{3}\right)^{-2}}{\frac{\left(\sqrt{3}\right)^{-2}}{\frac{1}{9}}}-\left(\log_{2}\left(32\right)\right)^{-1}+\frac{2}{7}\times \left(\frac{1}{7}\right)^{-1}$$

The base 2 logarithm of 32 is 5.

$$\frac{\left(\sqrt{3}\right)^{-2}}{\frac{\left(\sqrt{3}\right)^{-2}}{\frac{1}{9}}}-5^{-1}+\frac{2}{7}\times \left(\frac{1}{7}\right)^{-1}$$

Calculate 5 to the power of -1 and get \frac{1}{5}.

$$\frac{\left(\sqrt{3}\right)^{-2}}{\frac{\left(\sqrt{3}\right)^{-2}}{\frac{1}{9}}}-\frac{1}{5}+\frac{2}{7}\times \left(\frac{1}{7}\right)^{-1}$$

Calculate \frac{1}{7} to the power of -1 and get 7.

$$\frac{\left(\sqrt{3}\right)^{-2}}{\frac{\left(\sqrt{3}\right)^{-2}}{\frac{1}{9}}}-\frac{1}{5}+\frac{2}{7}\times 7$$

Multiply \frac{2}{7} and 7 to get 2.

$$\frac{\left(\sqrt{3}\right)^{-2}}{\frac{\left(\sqrt{3}\right)^{-2}}{\frac{1}{9}}}-\frac{1}{5}+2$$

Add -\frac{1}{5} and 2 to get \frac{9}{5}.

$$\frac{\left(\sqrt{3}\right)^{-2}}{\frac{\left(\sqrt{3}\right)^{-2}}{\frac{1}{9}}}+\frac{9}{5}$$

Divide \left(\sqrt{3}\right)^{-2} by \frac{\left(\sqrt{3}\right)^{-2}}{\frac{1}{9}} by multiplying \left(\sqrt{3}\right)^{-2} by the reciprocal of \frac{\left(\sqrt{3}\right)^{-2}}{\frac{1}{9}}.

$$\frac{\left(\sqrt{3}\right)^{-2}\times \frac{1}{9}}{\left(\sqrt{3}\right)^{-2}}+\frac{9}{5}$$

Cancel out \left(\sqrt{3}\right)^{-2} in both numerator and denominator.

$$\frac{1}{9}+\frac{9}{5}$$

Add \frac{1}{9} and \frac{9}{5} to get \frac{86}{45}.

$$\frac{86}{45}$$

Factor

$$\frac{2 \cdot 43}{3 ^ {2} \cdot 5} = 1\frac{41}{45} = 1.9111111111111112$$