Problem 16 - Algebra and Trigonometry

Question

Write an algebraic formula for the given quantity.. The sum \(S\) of the squares of two numbers \(n\) and \(m\)

Step-by-Step Solution

Answer

S = n^2 + m^2

Step 1: Understanding the Problem

We need to find an algebraic expression for the sum of the squares of two numbers, which are denoted as \(n\) and \(m\). This means we have to add the square of \(n\) and the square of \(m\) together.

Step 2: Writing the Expression for Each Square

For the number \(n\), its square is \(n^2\). Similarly, for the number \(m\), its square is \(m^2\).

Step 3: Adding the Squares Together

Now we sum the squares. The expression becomes \(n^2 + m^2\).

Step 4: Assigning the Sum to Variable S

Assign the sum of these squares to the variable \(S\). Thus, the formula is \(S = n^2 + m^2\).

Key Concepts

Sum of Squares

The concept of the sum of squares is frequently encountered in algebra. It involves adding the squares of numbers, which is a key operation in many mathematical areas.
The square of a number simply means multiplying the number by itself. For example, the square of a number \( n \) is represented as \( n^2 \), which stands for \( n \times n \). Similarly, the square of another number \( m \) would be \( m^2 \).
When you hear "sum of squares," it means you're going to add these squares together. So when we're talking about the sum of the squares of two numbers \( n \) and \( m \), it translates to the expression \( n^2 + m^2 \).
This formulation often emerges in geometry when finding the distance between points or in statistics when calculating variance or standard deviation.

Algebraic Formula

An algebraic formula is a way of expressing a mathematical relationship using symbols and variables. It provides a compact method to perform calculations and can be used to describe a wide range of scenarios.
In our example exercise, the formula \( S = n^2 + m^2 \) represents the sum of the squares of the numbers \( n \) and \( m \). The formula neatly condenses the process of squaring each number and then adding them together into one symbolic expression.
Recognizing and using algebraic formulas allows mathematicians to solve problems more efficiently. By understanding the structure of formulas, you can easily substitute variables and perform calculations quickly.

Variable Assignment

Variable assignment is a fundamental concept in algebra. It involves designating a specific letter or symbol to represent a value in an expression or equation. This process simplifies complex calculations by using shorthand.
In the exercise, we assign the symbol \( S \) to represent the sum of the squares of \( n \) and \( m \). This means every time we see \( S \), we know it refers to \( n^2 + m^2 \).
Variable assignment helps in organizing calculations and making equations easier to work with. It is like giving a name to a particular quantity, which can then be used in larger or more complex equations without repeatedly writing out the entire value.