Let's break down and solve the mathematical expressions in the provided image. The expressions are quite complex, so we will analyze and solve each part step by step.

Top Expression:

[ \frac{\log_{100}(10)}{\frac{1}{100}} + \sqrt[3]{\sqrt[3]{14661} \cdot \sqrt{16} \cdot \left(\sqrt{64} + \frac{16}{\sqrt{4}}\right)} ]

Middle Expression:

$\left(\frac{\left(\log_{100}(10)\right)^4}{\frac{1}{2} \cdot 16 \cdot \sin(30^\circ) \cdot \cos(60^\circ)}\right)^{-1} + \left(\frac{1}{2}\right)^{-1}$

Bottom Expression:

[ \sqrt[3]{\left(0.5\right)^{-1} \cdot \left(\sqrt[7]{(512 \cdot 0.5)^2}\right)^{10}} - 3 \cdot \left(6 \cdot \sin(30^\circ)\right)^3 + \frac{\sqrt[2]{(6 \cdot 0.5)^8}}{\cos(60^\circ)} + \frac{1}{21584 \cdot \cos(60^\circ)} ]

Now, let's start solving each part:

Top Expression:

1. $\log_{100}(10)$:

$\log_{100}(10) = \frac{\log_{10}(10)}{\log_{10}(100)} = \frac{1}{2}$

2. $\frac{1}{100}$:

$\frac{1}{100} = 0.01$

3. $\frac{\log_{100}(10)}{0.01}$:

$\frac{0.5}{0.01} = 50$

4. $\sqrt[3]{\sqrt[3]{14661} \cdot \sqrt{16} \cdot \left(\sqrt{64} + \frac{16}{\sqrt{4}}\right)}$:

• $\sqrt[3]{14661} \approx 24.43$
• $\sqrt{16} = 4$
• $\sqrt{64} = 8$
• $\frac{16}{\sqrt{4}} = 16 / 2 = 8$
• $\sqrt{64} + 8 = 8 + 8 = 16$

So,

[ \sqrt[3]{24.43 \cdot 4 \cdot 16} = \sqrt[3]{24.43 \cdot 64} \approx \sqrt[3]{1563.52} \approx 11.52 ]

Combining both parts:

$50 + 11.52 = 61.52$

Middle Expression:

1. $\left(\frac{\left(\log_{100}(10)\right)^4}{\frac{1}{2} \cdot 16 \cdot \sin(30^\circ) \cdot \cos(60^\circ)}\right)^{-1}$:

• $\log_{100}(10) = 0.5$
• $(0.5)^4 = 0.0625$
• $\sin(30^\circ) = 0.5$
• $\cos(60^\circ) = 0.5$
• $\frac{1}{2} \cdot 16 \cdot 0.5 \cdot 0.5 = 2$

So,

$\left(\frac{0.0625}{2}\right)^{-1} = \left(0.03125\right)^{-1} = 32$

2. $\left(\frac{1}{2}\right)^{-1}$:

$\left(\frac{1}{2}\right)^{-1} = 2$

Combining both parts:

$32 + 2 = 34$

Bottom Expression:

1. (\sqrt[3]{