Solve [(1/3)^{3}*(-1/3)^{5}*(-1/3)^2]^3:[-(-1/3)^7]^4

$$[ ( \frac { 1 } { 3 } ) ^ { 3 } \cdot ( - \frac { 1 } { 3 } ) ^ { 5 } \cdot ( - \frac { 1 } { 3 } ) ^ { 2 } ] ^ { 3 } : [ - ( - \frac { 1 } { 3 } ) ^ { 7 } ] ^ { 4 }$$

Evaluate

$$-\frac{1}{9}\approx -0.111111111$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 5 and 2 to get 7.

$$\frac{\left(\left(\frac{1}{3}\right)^{3}\left(-\frac{1}{3}\right)^{7}\right)^{3}}{\left(-\left(-\frac{1}{3}\right)^{7}\right)^{4}}$$

Calculate \frac{1}{3} to the power of 3 and get \frac{1}{27}.

$$\frac{\left(\frac{1}{27}\left(-\frac{1}{3}\right)^{7}\right)^{3}}{\left(-\left(-\frac{1}{3}\right)^{7}\right)^{4}}$$

Calculate -\frac{1}{3} to the power of 7 and get -\frac{1}{2187}.

$$\frac{\left(\frac{1}{27}\left(-\frac{1}{2187}\right)\right)^{3}}{\left(-\left(-\frac{1}{3}\right)^{7}\right)^{4}}$$

Multiply \frac{1}{27} and -\frac{1}{2187} to get -\frac{1}{59049}.

$$\frac{\left(-\frac{1}{59049}\right)^{3}}{\left(-\left(-\frac{1}{3}\right)^{7}\right)^{4}}$$

Calculate -\frac{1}{59049} to the power of 3 and get -\frac{1}{205891132094649}.

$$\frac{-\frac{1}{205891132094649}}{\left(-\left(-\frac{1}{3}\right)^{7}\right)^{4}}$$

Calculate -\frac{1}{3} to the power of 7 and get -\frac{1}{2187}.

$$\frac{-\frac{1}{205891132094649}}{\left(-\left(-\frac{1}{2187}\right)\right)^{4}}$$

The opposite of -\frac{1}{2187} is \frac{1}{2187}.

$$\frac{-\frac{1}{205891132094649}}{\left(\frac{1}{2187}\right)^{4}}$$

Calculate \frac{1}{2187} to the power of 4 and get \frac{1}{22876792454961}.

$$\frac{-\frac{1}{205891132094649}}{\frac{1}{22876792454961}}$$

Divide -\frac{1}{205891132094649} by \frac{1}{22876792454961} by multiplying -\frac{1}{205891132094649} by the reciprocal of \frac{1}{22876792454961}.

$$-\frac{1}{205891132094649}\times 22876792454961$$

Multiply -\frac{1}{205891132094649} and 22876792454961 to get -\frac{1}{9}.

$$-\frac{1}{9}$$

Factor

$$-\frac{1}{9} = -0.1111111111111111$$