Solve 8!/3!5!(2/3)^5(1/3)^3+0.259

$$\frac { 8 ! } { 3 ! 5 ! } ( \frac { 2 } { 3 } ) ^ { 5 } ( \frac { 1 } { 3 } ) ^ { 3 } + 0.259$$

Evaluate

$$\frac{3491299}{6561000}\approx 0.532129096$$

Solution Steps

The factorial of 8 is 40320.

$$\frac{40320}{3!\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}+0.259$$

The factorial of 3 is 6.

$$\frac{40320}{6\times 5!}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}+0.259$$

The factorial of 5 is 120.

$$\frac{40320}{6\times 120}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}+0.259$$

Multiply 6 and 120 to get 720.

$$\frac{40320}{720}\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}+0.259$$

Divide 40320 by 720 to get 56.

$$56\times \left(\frac{2}{3}\right)^{5}\times \left(\frac{1}{3}\right)^{3}+0.259$$

Calculate \frac{2}{3} to the power of 5 and get \frac{32}{243}.

$$56\times \frac{32}{243}\times \left(\frac{1}{3}\right)^{3}+0.259$$

Express 56\times \frac{32}{243} as a single fraction.

$$\frac{56\times 32}{243}\times \left(\frac{1}{3}\right)^{3}+0.259$$

Multiply 56 and 32 to get 1792.

$$\frac{1792}{243}\times \left(\frac{1}{3}\right)^{3}+0.259$$

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

$$\frac{1792}{243}\times \frac{1}{27}+0.259$$

Multiply \frac{1792}{243} times \frac{1}{27} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1792\times 1}{243\times 27}+0.259$$

Do the multiplications in the fraction \frac{1792\times 1}{243\times 27}.

$$\frac{1792}{6561}+0.259$$

Convert decimal number 0.259 to fraction \frac{259}{1000}.

$$\frac{1792}{6561}+\frac{259}{1000}$$

Least common multiple of 6561 and 1000 is 6561000. Convert \frac{1792}{6561} and \frac{259}{1000} to fractions with denominator 6561000.

$$\frac{1792000}{6561000}+\frac{1699299}{6561000}$$

Since \frac{1792000}{6561000} and \frac{1699299}{6561000} have the same denominator, add them by adding their numerators.

$$\frac{1792000+1699299}{6561000}$$

Add 1792000 and 1699299 to get 3491299.

$$\frac{3491299}{6561000}$$

Factor

$$\frac{43 \cdot 1657 \cdot 7 ^ {2}}{2 ^ {3} \cdot 3 ^ {8} \cdot 5 ^ {3}} = 0.5321290961743637$$