3.1 Simplify the following without the use of a calculator: 3.1.1 \( \frac{8 x^3 y^2}{16 x^4 y^4} \) (leave the answer with positive exponents) 3.1.2 \( \frac{\sqrt{48} + \sqrt{12}}{\sqrt{27}} \) 3.2 If \( \log 5 = m \), determine the following in terms of m: 3.2.1 \( \log 25 \) 3.2.2 \( \log_{2} (x + 3) - 3 = - \log_{2} (x - 4) \) 3.3 Solve for x: \( \log_{2} (x + 3) - 3 = - \log_{2} (x - 4) \) 3.4 Given complex numbers: \( z_1 = -1 + 3i \) and \( z_2 = \sqrt{2} \text{ cis } 135^{\circ} \) 3.4.1 Write down the conjugate of \( z_1 \) - NSC Technical Mathematics - Question 3 - 2022 - Paper 1

To simplify ( \frac{\sqrt{48} + \sqrt{12}}{\sqrt{27}} ):

1. Simplify ( \sqrt{48} = 4\sqrt{3} ) and ( \sqrt{12} = 2\sqrt{3} ) giving us: ( 4\sqrt{3} + 2\sqrt{3} = 6\sqrt{3} )
2. Simplify ( \sqrt{27} = 3\sqrt{3} )

The expression now is ( \frac{6\sqrt{3}}{3\sqrt{3}} = 2 ).

Using properties of logarithms:

( \log 25 = \log (5^2) = 2 \log 5 = 2m ).

Rearranging gives:

1. ( \log_{2} (x + 3) + \log_{2} (x - 4) = 3 )
2. Combine using the product property: ( \log_{2} ((x + 3)(x - 4)) = 3 )
3. Exponentiate: ( (x + 3)(x - 4) = 2^3 ) ( (x + 3)(x - 4) = 8 )
4. Expanding gives: ( x^2 - x - 12 = 8 ) leading to: ( x^2 - x - 20 = 0 ).

The conjugate of ( z_1 = -1 + 3i ) is ( z_1^* = -1 - 3i ).

Using the polar to rectangular conversion:

( z_2 = \sqrt{2} \text{ cis } 135^{\circ} = \sqrt{2} \left( \cos 135^{\circ} + i \sin 135^{\circ} \right) ) ( = \sqrt{2} \left( -\frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}} \right) = -1 + i ).

Using the obtained rectangular forms:

( z_1 \times z_2 = (-1 + 3i)(-1 + i) )

1. Expanding gives: ( = 1 - i - 3i + 3(-1) ) ( = 1 - 3 + (3 - 1)i = -2 + 2i ).

Rearranging gives:

( x + yi = 4 + 5i + 1 - i ) ( = 5 + 4i ).

Thus we have:

1. ( x = 5 )
2. ( y = 4 ).