Simplify (showing ALL calculations) the following without the use of a calculator: 3.1.1 \( \left( \frac{2a^{3}}{3} \right)^{3} \) 3.1.2 \( \log_{g} p + \log_{1} \) 3.1.3 \( \frac{\sqrt{48} - \sqrt{-12}}{2\sqrt{75}} \) - NSC Technical Mathematics - Question 3 - 2018 - Paper 1

Using the log property, we know that ( \log_{1} = 0 ), since any number to the power of 0 is 1.

Therefore, we can simplify:

$\log_{g} p + \log_{1} = \log_{g} p + 0 = \log_{g} p$

First, we simplify the numerator:

$\sqrt{48} - \sqrt{-12} = \sqrt{16 \cdot 3} - \sqrt{12} i = 4\sqrt{3} - 2\sqrt{3} i$

Now simplifying the denominator:

$2\sqrt{75} = 2\sqrt{25 \cdot 3} = 2(5\sqrt{3}) = 10\sqrt{3}$

Then the whole expression becomes:

$\frac{4\sqrt{3} - 2\sqrt{3}i}{10\sqrt{3}} = \frac{4 - 2i}{10} = \frac{2 - i}{5}$