Simplify the following expressions using exponent rules: a) 72 x (72)-2 x (2 x 7)23 x (22)-1 x 7⁻² b) (0.024 x 0.00057)/(0.95 x 0.0018 x 16)

42.

Step 1: Rewrite all numbers as powers of their prime factors.

$$\frac{7^2 \times (7^2)^{-2} \times (2 \times 7)}{2^3 \times (2^2)^{-1} \times 7^{-2}}$$

Step 2: Apply exponent rules $(a^m)^n = a^{mn}$ and $a^m \times a^n = a^{m+n}$.

$$\frac{7^2 \times 7^{-4} \times 2^1 \times 7^1}{2^3 \times 2^{-2} \times 7^{-2}}$$ $$\frac{2^1 \times 7^{2-4+1}}{2^{3-2} \times 7^{-2}}$$ $$\frac{2^1 \times 7^{-1}}{2^1 \times 7^{-2}}$$

Step 3: Apply exponent rule $\frac{a^m}{a^n} = a^{m-n}$.

$$2^{1-1} \times 7^{-1 - (-2)}$$ $$2^0 \times 7^{-1+2}$$ $$1 \times 7^1$$ $$7$$

The simplified value is $\boxed{7}$.

43.

Step 1: Convert decimals to fractions or powers of 10.

$$\frac{0.024 \times 0.00057}{0.95 \times 0.0018 \times 16} = \frac{\frac{24}{1000} \times \frac{57}{100000}}{\frac{95}{100} \times \frac{18}{10000} \times 16}$$ $$= \frac{24 \times 57 \times 100 \times 10000}{1000 \times 100000 \times 95 \times 18 \times 16}$$ $$= \frac{24 \times 57}{10 \times 10 \times 95 \times 18 \times 16}$$

Step 2: Simplify the numerical part.

$$= \frac{24 \times 57}{100 \times 95 \times 18 \times 16}$$

Divide 24 by 16: $24/16 = 3/2$.

$$= \frac{3 \times 57}{100 \times 95 \times 18 \times 2}$$

Divide 57 by 18: $57/18 = (3 \times 19)/(3 \times 6) = 19/6$.

$$= \frac{3 \times 19}{100 \times 95 \times 6 \times 2}$$

Divide 3 by 6: $3/6 = 1/2$.

$$= \frac{19}{100 \times 95 \times 2 \times 2}$$

Divide 19 by 95: $19/95 = 1/5$.

$$= \frac{1}{100 \times 5 \times 2 \times 2}$$ $$= \frac{1}{100 \times 20}$$ $$= \frac{1}{2000}$$

The value is $\boxed{\frac{1}{2000}}$.

44.

Step 1: A regular heptagon has 7 equal sides and 7 equal interior angles. It can be divided into 7 congruent isosceles triangles, with their vertices at the center O.

Step 2: Calculate the angle at the center for each triangle.

$$\theta = \frac{360^\circ}{7}$$

Step 3: The area of one such triangle (e.g., $\triangle OAB$) is given by the formula:

$$\text{Area of } \triangle OAB = \frac{1}{2} r^2 \sin(\theta)$$

Given radius $r = OB = 10 \text{ cm}$.

$$\text{Area of } \triangle OAB = \frac{1}{2} (10 \text{ cm})^2 \sin\left(\frac{360^\circ}{7}\right)$$ $$\text{Area of } \triangle OAB = \frac{1}{2} \times 100 \text{ cm}^2 \times \sin(51.42857^\circ)$$ $$\text{Area of } \triangle OAB = 50 \text{ cm}^2 \times 0.78183$$ $$\text{Area of } \triangle OAB \approx 39.0915 \text{ cm}^2$$

Step 4: Calculate the total area of the heptagon.

$$\text{Area of heptagon} = 7 \times \text{Area of } \triangle OAB$$ $$\text{Area of heptagon} = 7 \times 39.0915 \text{ cm}^2$$ $$\text{Area of heptagon} \approx 273.6405 \text{ cm}^2$$

Step 5: Round to 2 decimal places.

$$\text{Area of heptagon} \approx 273.64 \text{ cm}^2$$

The area of the heptagon is $\boxed{273.64 \text{ cm}^2}$.

45.