To simplify the expression (14 - \frac{13}{2} : \left(3\frac{11}{21} - 2\frac{7}{15}\right)), we will first convert the mixed numbers to improper fractions:
[14 - \frac{13}{2} : \left(3\frac{11}{21} - 2\frac{7}{15}\right)]
[= 14 - \frac{13}{2} : \left(\frac{64}{21} - \frac{37}{15}\right)]
Next, we convert the division into multiplication by taking the reciprocal of the divisor:
[= 14 - \frac{13}{2} \times \left(\frac{64}{21} - \frac{37}{15}\right)]
Now we multiply the fractions inside the parenthesis:
[= 14 - \frac{13}{2} \times \left(\frac{64}{21} - \frac{37}{15}\right)]
[= 14 - \frac{13}{2} \times \left(\frac{64 \times 15}{21 \times 15} - \frac{37 \times 21}{15 \times 21}\right)]
[= 14 - \frac{13}{2} \times \left(\frac{960}{315} - \frac{777}{315}\right)]
[= 14 - \frac{13}{2} \times \left(\frac{960 - 777}{315}\right)]
[= 14 - \frac{13}{2} \times \left(\frac{183}{315}\right)]
[= 14 - \frac{13 \times 183}{2 \times 315}]
[= 14 - \frac{2379}{630}]
Now we subtract the fractions:
[= \frac{8820 - 2379}{630}]
[= \frac{6441}{630}]
Now we can simplify the fraction by finding the greatest common divisor of 6441 and 630, which is 63:
[\frac{6441}{630} = \frac{6441 \div 63}{630 \div 63} = \frac{103}{10}]
Therefore, the simplified expression is (\frac{103}{10}).
