Write \(C++\) expressions for the following algebraic expressions: A) \(a=12 x\) B) \(z=5 x+14 y+6 k\) C) \(y=x^{4}\) D) \(g=\frac{h+12}{4 k}\) E) \(c=\frac{a^{3}}{b^{2} k^{4}}\)

C++ is a high-level programming language that allows for modular, procedural, and object-oriented programming. It is a popular choice for developing applications that require complex computations, like those seen in algebraic expressions. To translate an algebraic expression into a C++ expression, variables and operations must adhere to the syntax and structure of the language. For instance, multiplication is represented by an asterisk (*), and variables are typically declared with their type before being used, as seen in the translation of the algebraic expression 'a = 12x' to 'int a = 12 * x;' in C++.

In the given exercise, translating algebraic expressions into C++ requires an understanding of variable types. Whether we use 'int' for integers or 'double' for floating-point numbers will impact the accuracy and type of results we can expect. For example, using 'int' for the variable 'a' in the expression 'int a = 12 * x;' means that 'a' will only hold an integer value, even if 'x' results in a non-integer when multiplied by 12.

In programming, algebra is used to model and solve problems that deal with numbers and the relationships between them. Converting algebraic expressions into code involves translating the mathematical operations and logical structure into programming language constructs. While the exercise presents straightforward translations, understanding the underlying concepts is crucial.

For example, translating 'z = 5x + 14y + 6k' into 'int z = 5 * x + 14 * y + 6 * k;' reflects how the addition (+) and multiplication (*) operators work in C++. When faced with a division expression such as 'g = (h + 12) / (4k)', one must consider the data type to ensure proper accuracy and accommodate potential decimal results. This is why 'double g = static_cast(h + 12) / (4 * k);', with a cast to double, is used.

This consideration is part of what makes translating algebraic expressions to code both an exact and nuanced task, requiring attention to the implications of each algebraic operation.

The cmath library in C++ contains a wide range of mathematical functions that are useful when dealing with algebraic expressions. It provides functions such as 'pow()', which raises a number to a certain power, and is essential in translating expressions like 'y = x^4' into C++ code, 'int y = pow(x, 4);'. The pow() function takes two arguments: the base and the exponent, and it returns the base raised to the power of the exponent.

careful use of these functions is important when translating expressions that involve non-integer powers or when the result may not be an integer. In the exercise, the use of 'pow()' is demonstrated in the translation of 'c = (a^3) / (b^2 * k^4)' to 'double c = pow(a, 3) / (pow(b, 2) * pow(k, 4));', showcasing how the cmath library plays a crucial role in implementing complex algebraic expressions in C++. Including this library by adding '#include ' at the beginning of a C++ program is necessary to access these powerful mathematical functions and perform precise computations.