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Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It allows us to create models of real-world situations and solve problems using equations. In the context of our exercise, algebra helps us evaluate the function \( g(s, t) = 3s\sqrt{t} + t\sqrt{s} + 2 \) by substituting specific values for \( s \) and \( t \).

• **Variables**: In our function, the symbols \( s \) and \( t \) represent variables. Variables are placeholders that can change and allow us to generalize mathematical expressions.
• **Substitution**: By plugging different numbers into the variables, we can determine specific outcomes, as shown in our solution steps.
• **Operations**: Algebra involves various operations like addition, multiplication, and in this case, working with square roots, which are crucial for solving these expressions.

Understanding algebra is essential for evaluating functions, as it provides the structure needed to manipulate and solve expressions correctly.

A mathematical expression is a combination of numbers, variables, and operations. It's like a recipe that tells us how to calculate a specific value. Evaluating mathematical expressions involves substituting variables with actual numbers and performing operations according to established rules.
- **Components**: In our expression \( g(s, t) \), we have parts like \( 3s\sqrt{t} \), \( t\sqrt{s} \), and a constant part \(+ 2\). Each of these components needs to be handled according to mathematical operations.- **Order of Operations**: Remember, when evaluating expressions, follow the order of operations (BODMAS/PEDMAS): brackets, orders (like square roots), division, multiplication, addition, and subtraction.- **Example**: For \( g(1, 2) \), we substitute \( s = 1 \) and \( t = 2 \) and follow the order of operations to find \( 3\sqrt{2} + 4 \).
By carefully following these steps, you can evaluate any expression correctly.

The square root is a special number that, when multiplied by itself, gives the original number. Symbolically, the square root of \( x \) is denoted as \( \sqrt{x} \). In the exercise, square roots are crucial for evaluating the function \( g(s, t) \).
- **Definition**: If \( y^2 = x \), then \( y \) is the square root of \( x \). For example, since \( 3^2 = 9 \), \( 3 \) is the square root of \( 9 \).- **Rational vs. Irrational**: Some square roots result in whole numbers, like \( \sqrt{4} = 2 \), while others, like \( \sqrt{2} \), are irrational numbers, meaning they cannot be precisely expressed as a simple fraction.- **Simplifying Expressions**: When simplifying expressions that include square roots, they must be carefully calculated, as with expressions like \( 3s\sqrt{t} \). This requires squaring or square rooting the correct terms.
Understanding square roots is essential as it frequently appears in various mathematical operations and significantly impacts how we evaluate expressions.