Simplify the following expressions: a) 16^ rac12 imes 27^ rac13 b) 8^ rac53 imes 49^ rac32. Use logarithms to work out: a) 3/(8.237 x 0.0045)

$16^{\frac{1}{2}} = \sqrt{16} = 4$  
$27^{\frac{1}{3}} = \sqrt[3]{27} = 3$

$4 \times 3 = 12$

The simplified expression is $\boxed{12}$.

$8^{\frac{5}{3}} = (\sqrt[3]{8})^5 = (2)^5 = 32$  
$49^{\frac{3}{2}} = (\sqrt{49})^3 = (7)^3 = 343$

$32 \times 343 = 10976$

The simplified expression is $\boxed{10976}$.

$\log(21.8) = 1.3385$  
$\log(0.1275) = \bar{1}.1055$  
$\log(8.237) = 0.9158$  
$\log(0.0045) = \bar{3}.6532$

$\log(\text{numerator}) = \log(21.8) + \log(0.1275)$  
$= 1.3385 + \bar{1}.1055$  
$= (1 + \bar{1}) + (0.3385 + 0.1055)$  
$= 0 + 0.4440 = 0.4440$

$\log(\text{denominator}) = \log(8.237) + \log(0.0045)$  
$= 0.9158 + \bar{3}.6532$  
$= (0 + \bar{3}) + (0.9158 + 0.6532)$  
$= -3 + 1.5690 = -2 + 0.5690 = \bar{2}.5690$

$\log\left(\frac{\text{numerator}}{\text{denominator}}\right) = \log(\text{numerator}) - \log(\text{denominator})$  
$= 0.4440 - \bar{2}.5690$  
$= 0.4440 - (-2 + 0.5690)$  
$= 0.4440 + 2 - 0.5690$  
$= 2.4440 - 0.5690 = 1.8750$

$\log X = \frac{1}{3} (1.8750) = 0.6250$

$X = \text{antilog}(0.6250) \approx 4.217$

The value is $\boxed{4.217}$.

$\log(0.4583) = \bar{1}.6612$  
$\log(2387) = 3.3778$  
$\log(2.883) = 0.4598$

$\log(\text{denominator}) = \log(2387) + \log(2.883)$  
$= 3.3778 + 0.4598 = 3.8376$

$\log\left(\frac{\text{numerator}}{\text{denominator}}\right) = \log(0.4583) - \log(\text{denominator})$  
$= \bar{1}.6612 - 3.8376$  
$= (-1 + 0.6612) - 3.8376$  
$= -1 - 3.8376 + 0.6612$  
$= -4.8376 + 0.6612 = -4.1764$

To make the characteristic divisible by 5, we write $-4.1764$ as $-5 + 0.8236 = \bar{5}.8236$.