Simplify the following decimal fractions to the form a/b where a; b ∈ Z and b ≠ 0. Leave the answer in their simplest form.

$$= \frac{58 \div 2}{100 \div 2} = \frac{29}{50}$$

The simplified fraction is $\boxed{\mathbf{\frac{29}{50}}}$.

1.1.2 2.14

$$2 + \frac{7}{50} = \frac{100}{50} + \frac{7}{50} = \frac{107}{50}$$

The simplified fraction is $\boxed{\mathbf{\frac{107}{50}}}$.

1.1.3 0.375

$$\frac{375 \div 125}{1000 \div 125} = \frac{3}{8}$$

The simplified fraction is $\boxed{\mathbf{\frac{3}{8}}}$.

1.2 Simplify the following by using the laws of exponents and leave answers with positive exponents and in surd form where applicable:

1.2.1 $\left(\frac{81a^3}{16a^{-1}}\right)^{-\frac{3}{2}}$

$$= \frac{4^3}{(9a^2)^3} = \frac{64}{9^3 (a^2)^3} = \frac{64}{729a^6}$$

The simplified expression is $\boxed{\mathbf{\frac{64}{729a^6}}}$.

1.2.2 $\frac{x^2y^3 \times x^3y^4}{(2x^{-3}y)^2} + \sqrt{4x^2y^3}$

Assume $x \ge 0$ for $\sqrt{x^2}=x$.

$$\sqrt{4x^2y^3} = \sqrt{4} \cdot \sqrt{x^2} \cdot \sqrt{y^2 \cdot y} = 2xy\sqrt{y}$$

$$\frac{1}{4} x^{11} y^5 + 2xy\sqrt{y}$$

The simplified expression is $\boxed{\mathbf{\frac{1}{4} x^{11} y^5 + 2xy\sqrt{y}}}$.

1.3 Simplify the following by using surd laws. Leave answer in a surd form.

$$\frac{\sqrt{75}+\sqrt{12}+\sqrt{108}}{\sqrt{147}}$$

$$\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}$$ $$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$$ $$\sqrt{108} = \sqrt{36 \times 3} = 6\sqrt{3}$$

$$= \frac{13}{7}$$

The simplified expression is $\boxed{\mathbf{\frac{13}{7}}}$.

1.4 Simplify the following expression by rationalising the denominator.

$$\frac{2-\sqrt{3}}{3-\sqrt{2}}$$

$$(2-\sqrt{3})(3+\sqrt{2}) = 2(3) + 2(\sqrt{2}) - \sqrt{3}(3) - \sqrt{3}(\sqrt{2})$$ $$= 6 + 2\sqrt{2} - 3\sqrt{3} - \sqrt{6}$$

$$= \frac{6 + 2\sqrt{2} - 3\sqrt{3} - \sqrt{6}}{7}$$

The simplified expression is $\boxed{\mathbf{\frac{6 + 2\sqrt{2} - 3\sqrt{3} - \sqrt{6}}{7}}}$.

1.5 Given: $A_t = A_0 \left(1 + \frac{r}{100}\right)^n$

1.5.1 Make "r" the subject of the formula.

$$r = 100 \left(\sqrt[n]{\frac{A_t}{A_0}} - 1\right)$$

The formula with $r$ as the subject is $\boxed{\mathbf{r = 100 \left(\sqrt[n]{\frac{A_t}{A_0}} - 1\right)}}$.

1.5.2 Determine the value of r if n=4, $A_0=3000$ and $A_t=4200$.

$$r = 100 \left(\sqrt[4]{\frac{42}{30}} - 1\right) = 100 \left(\sqrt[4]{\frac{7}{5}} - 1\right)$$ $$r = 100 \left(\sqrt[4]{1.4} - 1\right)$$

$$r \approx 100 (1.08775 - 1)$$ $$r \approx 100 (0.08775)$$ $$r \approx 8.775$$

The value of $r$ is approximately $\boxed{\mathbf{8.775}}$.

(Question 1.6 was addressed in a previous message.)