https://file.aiquickdraw.com/mathbot/file/17301427861515r3jqoe1.jpg

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]{

Solve the given mathematical expressions shown in the image.

Explore the solution to a complex mathematical expression involving logarithmic calculations, root extractions, exponent manipulations, and trigonometric functions. This problem is suitable for students in Grades 10-12 and provides step-by-step breakdowns of each component.

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