Problem 83 - Algebra A Combined Function

Question

Fill in the boxes so that the terms in each expression can be combined. Then simplify. Each exercise has more than one solution. $$ 7 x^{\square}+2 x^{\square} $$

Step-by-Step Solution

Answer

The terms combine to form \(9x^{\square}\).

Step 1: Identify the Like Terms

In the expression \(7x^{\square} + 2x^{\square}\), the terms \(7x^{\square}\) and \(2x^{\square}\) are like terms. This means the base and exponent are the same in each term.

Step 2: Define the Exponent

Since both terms are like terms, they must have the same exponent. Choose an integer for the exponent of the terms, such as \(n\) and redefine the expression as \(7x^n + 2x^n\) where \(n\) represents any whole number.

Step 3: Combine Like Terms

Once the terms have the same exponent, add the coefficients. Combine \(7x^n\) and \(2x^n\) to get \((7 + 2)x^n = 9x^n\).

Step 4: Simplify the Expression

The combined expression is \(9x^n\). This is the simplified form of the original expression when the exponent \(n\) is the same for both terms.

Key Concepts

Algebraic Expressions

In algebra, expressions consist of numbers, variables, and operators. These components come together to form terms. A classic example is the expression \(7x^{\square}+2x^{\square}\). Here, \(x\) represents a variable that's raised to a power indicated by the exponent. Variables can symbolize different quantities. Operators like addition or subtraction connect these terms within the expression.
An algebraic expression doesn't rely solely on numbers, making it versatile in representing various scenarios. This flexibility is what makes expressions a cornerstone in algebra. They help us describe patterns, solve problems, and explore relationships between variables and constants.

Simplifying Expressions

The process of simplifying expressions involves reducing an expression to its simplest form. This means combining like terms and removing any unnecessary complexity. In our example \(7x^{n} + 2x^{n}\), simplifying involves identifying terms that can be combined based on their similarity by ensuring they have the same variables raised to the same power.
By adding the coefficients of these terms, we condense the expression to \(9x^{n}\). Simplifying expressions helps reduce the amount of work needed in algebra, making them easier to handle in equations and word problems. Simplifying is essential for clear, concise, and accurate solving of algebraic problems.

Exponents

Exponents play a crucial role in algebraic expressions. They indicate how many times a variable is multiplied by itself. For example, in \(x^{n}\), the exponent \(n\) tells us that \(x\) is used as a factor \(n\) times. Exponents allow us to express large numbers succinctly and perform rapid calculations.
When combining like terms with exponents, the exponents must be the same for addition or subtraction to be possible, as seen with \(7x^n\) and \(2x^n\). Since the exponents match, we can combine their coefficients. Understanding exponents helps in grasping more complex algebraic concepts and aids in simplifying expressions efficiently.

Coefficients

Coefficients in algebra are the numerical factors in terms. In the expression \(7x^{\square} + 2x^{\square}\), the numbers 7 and 2 are coefficients. They indicate how many times a term is counted. Coefficients are crucial when combining like terms because they are the sum or difference used to form a simplified expression.
In our example, when the terms \(7x^n\) and \(2x^n\) are combined, the coefficients 7 and 2 are added together to form the new coefficient 9, resulting in \(9x^n\). Understanding coefficients is essential in manipulating expressions, allowing us to efficiently add, subtract, and factor expressions.