Apply surd rules to simplify the following without the use of a calculator and give the answer with a rational denominator. Show ALL steps: 1.3.1 \( \frac{\sqrt{10}-\sqrt{5}}{\sqrt{10}} \) \( 1.3 .2 \frac{\sqrt{48 x} x+\sqrt{\left(27 x^{2}\right)}}{\sqrt{\left(48 x^{2}\right)}} \) Given \( A_{t}=A_{0}(1+r t) \) 1.4.1 Make \( t \) the subject of the formula.

To simplify \( \frac{\sqrt{10}-\sqrt{5}}{\sqrt{10}} \), we can first rewrite it by separating the fraction: \[ \frac{\sqrt{10}}{\sqrt{10}} - \frac{\sqrt{5}}{\sqrt{10}} = 1 - \frac{\sqrt{5}}{\sqrt{10}} \] Now, we need to rationalize the denominator of the second term: \[ \frac{\sqrt{5}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}} = \frac{\sqrt{50}}{10} = \frac{5\sqrt{2}}{10} = \frac{\sqrt{2}}{2} \] So, the expression simplifies to: \[ 1 - \frac{\sqrt{2}}{2} = \frac{2}{2} - \frac{\sqrt{2}}{2} = \frac{2 - \sqrt{2}}{2}. \] --- For the second expression \( \frac{\sqrt{48x}x + \sqrt{27x^2}}{\sqrt{48x^2}} \), we start by simplifying the numerator. \[ \sqrt{48x}x + \sqrt{27x^2} = x\sqrt{48x} + 3x\sqrt{3} = x(\sqrt{48x} + 3\sqrt{3}). \] Now, since \( \sqrt{48} = 4\sqrt{3} \), we can rewrite that as: \[ \sqrt{48x} = 4\sqrt{3}\sqrt{x}, \] So the numerator becomes: \[ x(4\sqrt{3}\sqrt{x} + 3\sqrt{3}) = x\sqrt{3}(4\sqrt{x} + 3). \] Now we can simplify the denominator: \[ \sqrt{48x^2} = \sqrt{48}\sqrt{x^2} = 4\sqrt{3}x. \] Putting it all together, we have: \[ \frac{x\sqrt{3}(4\sqrt{x} + 3)}{4\sqrt{3}x}. \] Cancelling out \( x \) (assuming \( x \neq 0 \)), we get: \[ \frac{4\sqrt{x} + 3}{4}. \] --- To make \( t \) the subject of the formula \( A_t = A_0(1 + rt) \), start by isolating the term involving \( t \): \[ \frac{A_t}{A_0} = 1 + rt. \] Next, subtract 1 from both sides: \[ \frac{A_t}{A_0} - 1 = rt. \] Then, multiply both sides by \( \frac{1}{r} \): \[ t = \frac{\frac{A_t}{A_0} - 1}{r}. \] This shows that \( t \) is expressed in terms of \( A_t \), \( A_0 \), and \( r \).