3.1 Vereenvoudig die volgende, sonder om 'n sakrekenaar te gebruik: 3.1.1 7(3x)⁰ 3.1.2 \(\sqrt{\left(242 - \sqrt{72}\right)}\) 3.1.3 \(\frac{9^{1 - x} \cdot 27^{2 - n}}{81^{2 - n}}\) 3.2 Los op vir x: \(\log(x + 2) - \log x = 1\) 3.3 Gee die komplekse getal: \(z = 5 - 5i\) 3.3.1 Skryf die kwadrant in die kompleksvlak neer waarin z lê - NSC Technical Mathematics - Question 3 - 2023 - Paper 1

To simplify this expression, we first find (\sqrt{72}):

$\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$

Thus, we can rewrite the expression:

$\sqrt{242 - 6\sqrt{2}}$

Next, we need to find (242) as an easier term:

$\sqrt{242} \approx 15.556$

This gives us:

$\sqrt{242 - 6\sqrt{2}}$

Therefore, we find that the result approximates or represents a number around 20.

We can rewrite the bases as powers of 3:

$9 = 3^2, \quad 27 = 3^3, \quad 81 = 3^4$

Substituting these values gives:

$\frac{(3^2)^{1 - x} \cdot (3^3)^{2 - n}}{(3^4)^{2 - n}} = \frac{3^{2(1 - x)} \cdot 3^{3(2 - n)}}{3^{4(2 - n)}}$

Using properties of exponents:

$3^{2 - 2x + 6 - 3n - 8 + 4n} = 3^{-2 + n - 2x}$

So the result simplifies to:

$3^{-2 + n - 2x}$

Using the properties of logarithms, we can combine the logs:

$\log\left(\frac{x + 2}{x}\right) = 1$

This implies:

$\frac{x + 2}{x} = 10$

Multiplying through by x yields:

$x + 2 = 10x$

Rearranging gives:

$2 = 9x \quad \Rightarrow \quad x = \frac{2}{9}$

The complex number is (z = 5 - 5i), which lies in the fourth quadrant since the real part (5) is positive and the imaginary part (-5) is negative.

To express (z) in polar form, we calculate the modulus and argument:

1. Modulus:

$r = \sqrt{5^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}$

2. Argument:

Using (\tan\theta = \frac{\text{Im}}{\text{Re}} = \frac{-5}{5} = -1 \Rightarrow \theta = -\frac{\pi}{4}$$

Thus, the polar form is:

$z = r(\cos(\theta) + i\sin(\theta)) = 5\sqrt{2}\left(\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4})\right)$

First, expand the expression:

$m = 3i(2i) - 3i(5) + 7 - ni = 6i^2 - 15i + 7 - ni$

Since (i^2 = -1), substitute this to find:

$m = 6(-1) - 15i + 7 - ni = -6 - 15i + 7 - ni = 1 - (15 + n)i$

Thus, separating real and imaginary parts gives:

Real part: 1, Imaginary part: (-(15 + n))