Simplify the following expressions: i) 28x11 / 7x9 ii) 25m12 / 5m7. Evaluate P = sqrt((269.1 x 89.33)/(41.72)) using logarithms.

The expression is $28x^{11} \div 7x^9$.
To simplify, divide the coefficients and use the exponent rule $a^m \div a^n = a^{m-n}$ for the variables.

$$\frac{28x^{11}}{7x^9} = \left(\frac{28}{7}\right) \times \left(\frac{x^{11}}{x^9}\right)$$ $$= 4 \times x^{(11-9)}$$ $$= 4x^2$$

The simplified expression is $\boxed{4x^2}$.

The expression is $25m^{12} \div 5m^7$.
To simplify, divide the coefficients and use the exponent rule $a^m \div a^n = a^{m-n}$ for the variables.

$$\frac{25m^{12}}{5m^7} = \left(\frac{25}{5}\right) \times \left(\frac{m^{12}}{m^7}\right)$$ $$= 5 \times m^{(12-7)}$$ $$= 5m^5$$

The simplified expression is $\boxed{5m^5}$.

Let $P = \sqrt{\frac{269.1 \times 89.33}{41.72}}$.
This can be written as $P = \left(\frac{269.1 \times 89.33}{41.72}\right)^{\frac{1}{2}}$.
Take the logarithm (base 10) of both sides:

$$\log P = \log \left[\left(\frac{269.1 \times 89.33}{41.72}\right)^{\frac{1}{2}}\right]$$

Using the logarithm property $\log(a^b) = b \log a$:

$$\log P = \frac{1}{2} \log \left(\frac{269.1 \times 89.33}{41.72}\right)$$

Using the logarithm properties $\log(\frac{a}{b}) = \log a - \log b$ and $\log(ab) = \log a + \log b$:

$$\log P = \frac{1}{2} [\log(269.1) + \log(89.33) - \log(41.72)]$$

Now, find the logarithms of each number:

$$\log(269.1) \approx 2.4300$$ $$\log(89.33) \approx 1.9519$$ $$\log(41.72) \approx 1.6203$$

Substitute these values into the equation for $\log P$:

$$\log P = \frac{1}{2} [2.4300 + 1.9519 - 1.6203]$$ $$\log P = \frac{1}{2} [4.3819 - 1.6203]$$ $$\log P = \frac{1}{2} [2.7616]$$ $$\log P = 1.3808$$

Finally, find the antilog of 1.3808 to get $P$:

$$P = 10^{1.3808}$$ $$P \approx 24.03$$

The value of the expression is $\boxed{24.03}$.