Vereenvoudig die volgende SONDERS die gebruik van 'n sakrekenaar: 3.1.1 \(rac{(81a^2)}{(3^4)}\) $^{\frac{3}{4}}$ 3.1.2 \(log_2 16 + log_4 4\) 3.1.3 \(\sqrt{50x^{10} \times 18x^{4}}\) Los op vir \(x:\) \(log_2 (x + 2) = 2 + log_3 x\) Gegge die komplekste getal \(z = p + 4i\) met die modulus \(\frac{\sqrt{5}}{2}\) en die argument \(\theta = (90^\circ ; 180^\circ)\) 3.3.1 Bepaal die waarde van \(p\) - NSC Technical Mathematics - Question 3 - 2021 - Paper 1

We can utilize the properties of logarithms:

1. Start with (log_2 16): Since (16 = 2^4), we have: [ log_2 16 = 4 ]

2. Next, consider (log_4 4): Since it's the same base and number: [ log_4 4 = 1 ]

3. Finally, add these results together: [ 4 + 1 = 5 ]

Thus, the final answer is (5).

To simplify this expression, we'll break it down step by step:

1. First, calculate the constants: [ 50 \times 18 = 900 ]

2. Then, handle the variables: [ x^{10} \times x^{4} = x^{14} ]

3. Now, we can bring it all together inside the square root: [ \sqrt{900x^{14}} = \sqrt{900} \times \sqrt{x^{14}} = 30 \times x^{7} ]

Thus, the simplified form is (30x^{7}).

We will solve this equation step by step:

1. First, rewrite the equation: [ log_2 (x + 2) = log_3 x + 2 ]

2. Convert (2) into logarithm form: [ 2 = log_2 4 \implies log_2 (x + 2) = log_3 x + log_2 4 ]

3. Combine the logarithms on the right: [ log_2 (x + 2) = log_3 (4x) ]

4. Since we have the logarithm bases different, we'll take an exponential form to solve: [ x + 2 = \frac{4x}{\log_3 2} ]

5. Setting up the equation for solutions leads to: [ x + 2 = 4x \cdot \frac{1}{\log_3 2} ]

6. We find that: [ \Rightarrow x = \frac{2}{3 \log_3 2 - 1} ] Solving for (x) yields: (x = 0.25).

Given the complex number: ( z = p + 4i ) and the modulus: ( |z| = \frac{\sqrt{5}}{2}), we will solve for (p):

1. Apply the modulus formula: [ |z| = \sqrt{p^2 + 4^2} ightarrow \sqrt{p^2 + 16} = \frac{\sqrt{5}}{2} ]

2. Square both sides: [ p^2 + 16 = \frac{5}{4} ightarrow p^2 = \frac{5}{4} - 16 = \frac{5 - 64}{4} ightarrow p^2 = \frac{-59}{4} ]

Thus, this shows (p = \sqrt{-59/4}), which means (p = -2) or (p = 2i).

To express (z) in polar form:

1. Calculate the modulus: [ r = |z| = \sqrt{p^2 + 16} ]

2. Calculate the angle (\theta) using the known values from previous sections. Also, note that the modulus and argument: [ \theta = tan^{-1}\left(\frac{4}{0}\right) = \frac{\pi}{2} ]

3. Thus, we can express it as: [ z = r \text{cis} \theta = r(\cos \theta + i\sin \theta) ]

Combining these forms gives you the polar form of the complex number.

To solve for (m) and (n):

1. Start by simplifying the right side: [ -3i(4 + 5) = -27i ]

2. Set up the equation: [ 2m - ni - 6i = -27i ]

3. Equate real and imaginary parts: [ 2m = 0 \quad and \quad -n - 6 = -27 ]

4. Solve for (m): [ m = 0 \quad \Rightarrow 2m = 12 ]

5. Solve for (n): [ -n - 6 = -27 \Rightarrow -n = -21 \Rightarrow n = 21 ]

Thus, the final answer gives (m = 0) and (n = 21).