Problem 29 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 14 a^{2} b+4 a^{2} b+19 a^{2} b $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression \(14a^2b + 4a^2b + 19a^2b\). Answer: \(37a^2b\)

Step 1: Identify like terms

In the given expression, all three terms are like terms because they have the same variable and exponent, \(a^2\) and \(b\). These terms are: \(14a^2b\), \(4a^2b\), and \(19a^2b\).

Step 2: Combine like terms

To combine the like terms from Step 1, we will add the coefficients of each term: \(14 + 4 + 19\).

Step 3: Perform the addition

Add the coefficients from Step 2: \(14 + 4 + 19 = 37\).

Step 4: Write the simplified expression

Now that we have the combined coefficient, we can write the simplified expression by attaching the common variable and exponent, \(a^2\) and \(b\), to the combined coefficient: \(37a^2b\). The simplified expression is: $$ 37 a^{2} b $$

Key Concepts

Combining Like Terms

Simplifying algebraic expressions often involves combining like terms, which are terms that share the same variables and exponents. This process consolidates the expression into fewer terms, making it easier to manage and solve. For example, in the expression 14a^2b + 4a^2b + 19a^2b, all the terms are like terms because they have exactly the same variables raised to the same powers - in this case, a^2 and b.

To combine like terms:

• First, group all the like terms together.
• Second, add or subtract their numerical coefficients.
• Finally, write the resulting coefficient next to the common variable part.

Using these steps turns the original expression into 37a^2b, which is the simplified version of the algebraic expression.

Algebraic Expression

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operations (such as addition and multiplication). The key feature of an algebraic expression is that it represents a value. For our exercise, 14a^2b + 4a^2b + 19a^2b is an algebraic expression consisting of three terms.

Algebraic expressions are essential in forming equations and solving problems across various fields of mathematics. It is important for students to become comfortable in working with these expressions, as they are foundational to understanding more complex mathematical concepts.

Coefficients

Coefficients are the numerical factor of a term with a variable. In algebra, they are the numbers you multiply a variable or variables by. If you have an expression like 5x, the number 5 is the coefficient. In the expression from the exercise, 14a^2b, 4a^2b, and 19a^2b each have coefficients of 14, 4, and 19 respectively.

When simplifying expressions by combining like terms, you mainly deal with coefficients, adding or subtracting them as required. It's crucial to understand that only the coefficients are combined; the variable part does not change when simplifying expressions by combining like terms.

Exponents

Exponents are used to represent repeated multiplication of a number by itself. In algebra, the exponent is the small number placed above and to the right of a variable or number, also known as a power. For instance, a^2 indicates that a is to be multiplied by itself. In our example expression, the term a^2b contains the variable a with an exponent of 2, indicating a * a.

Understanding the role of exponents is crucial when combining like terms since only those terms with identical variable combinations and exactly the same exponents can be combined. Different exponents would indicate that the terms represent different quantities and can't be combined by simple arithmetic operations.