How would you use Lua's mathematical functions to perform calculations? Illustrate with examples of common mathematical operations such as addition, subtraction, and exponentiation.

Lua provides a set of mathematical functions that enable you to perform various calculations. These functions are part of the standard math library in Lua. Here's an in-depth explanation of how you can use Lua's mathematical functions to perform common mathematical operations:

1. Addition:
The addition operation can be performed using the `+` operator. Lua's mathematical functions are not directly involved in simple addition as they are primarily designed for more complex operations. Here's an example:

```
lua`local result = 10 + 5
print(result) -- Output: 15`
```
In this example, the `+` operator is used to add the numbers 10 and 5, resulting in 15.
2. Subtraction:
Similar to addition, subtraction can be performed using the `-` operator. Here's an example:

```
lua`local result = 20 - 8
print(result) -- Output: 12`
```
In this example, the `-` operator is used to subtract 8 from 20, resulting in 12.
3. Exponentiation:
Lua provides the `math.pow()` function to calculate exponentiation. The `math.pow()` function takes two arguments: the base and the exponent. Here's an example:

```
lua`local result = math.pow(2, 3)
print(result) -- Output: 8`
```
In this example, `math.pow()` is used to calculate 2 raised to the power of 3, resulting in 8.
4. Other Mathematical Functions:
Lua's math library provides various other mathematical functions that enable you to perform advanced calculations. Some commonly used functions include `math.sin()` for calculating the sine of an angle, `math.cos()` for calculating the cosine, `math.sqrt()` for calculating the square root, and `math.abs()` for finding the absolute value of a number.

Example:

```
lua`local angle = math.pi / 4
local sinValue = math.sin(angle)
print(sinValue) -- Output: 0.70710678118655

local value = -10
local absoluteValue = math.abs(value)
print(absoluteValue) -- Output: 10`
```
In this example, `math.sin()` is used to calculate the sine of an angle, and `math.abs()` is used to find the absolute value of a number.

By utilizing Lua's mathematical functions and operators, you can perform various calculations, including addition, subtraction, exponentiation, and more. These functions provide a comprehensive set of tools for performing mathematical operations in Lua, allowing you to manipulate numerical data effectively in your programs.