Solve 27[1-(-2/3)^18/1-(-frac{2{3})}]

$$27 [ \frac { 1 - ( - \frac { 2 } { 3 } ) ^ { 18 } } { 1 - ( - \frac { 2 } { 3 } ) } ]$$

Evaluate

$$\frac{77431669}{4782969}\approx 16.18903844$$

Solution Steps

Calculate -\frac{2}{3} to the power of 18 and get \frac{262144}{387420489}.

$$27\times \frac{1-\frac{262144}{387420489}}{1-\left(-\frac{2}{3}\right)}$$

Convert 1 to fraction \frac{387420489}{387420489}.

$$27\times \frac{\frac{387420489}{387420489}-\frac{262144}{387420489}}{1-\left(-\frac{2}{3}\right)}$$

Since \frac{387420489}{387420489} and \frac{262144}{387420489} have the same denominator, subtract them by subtracting their numerators.

$$27\times \frac{\frac{387420489-262144}{387420489}}{1-\left(-\frac{2}{3}\right)}$$

Subtract 262144 from 387420489 to get 387158345.

$$27\times \frac{\frac{387158345}{387420489}}{1-\left(-\frac{2}{3}\right)}$$

The opposite of -\frac{2}{3} is \frac{2}{3}.

$$27\times \frac{\frac{387158345}{387420489}}{1+\frac{2}{3}}$$

Convert 1 to fraction \frac{3}{3}.

$$27\times \frac{\frac{387158345}{387420489}}{\frac{3}{3}+\frac{2}{3}}$$

Since \frac{3}{3} and \frac{2}{3} have the same denominator, add them by adding their numerators.

$$27\times \frac{\frac{387158345}{387420489}}{\frac{3+2}{3}}$$

Add 3 and 2 to get 5.

$$27\times \frac{\frac{387158345}{387420489}}{\frac{5}{3}}$$

Divide \frac{387158345}{387420489} by \frac{5}{3} by multiplying \frac{387158345}{387420489} by the reciprocal of \frac{5}{3}.

$$27\times \frac{387158345}{387420489}\times \frac{3}{5}$$

Multiply \frac{387158345}{387420489} times \frac{3}{5} by multiplying numerator times numerator and denominator times denominator.

$$27\times \frac{387158345\times 3}{387420489\times 5}$$

Do the multiplications in the fraction \frac{387158345\times 3}{387420489\times 5}.

$$27\times \frac{1161475035}{1937102445}$$

Reduce the fraction \frac{1161475035}{1937102445} to lowest terms by extracting and canceling out 15.

$$27\times \frac{77431669}{129140163}$$

Express 27\times \frac{77431669}{129140163} as a single fraction.

$$\frac{27\times 77431669}{129140163}$$

Multiply 27 and 77431669 to get 2090655063.

$$\frac{2090655063}{129140163}$$

Reduce the fraction \frac{2090655063}{129140163} to lowest terms by extracting and canceling out 27.

$$\frac{77431669}{4782969}$$

Factor

$$\frac{7 \cdot 19 \cdot 577 \cdot 1009}{3 ^ {14}} = 16\frac{904165}{4782969} = 16.1890384403495$$