Question

Combine as indicated and simplify. $$8.5 a b-21.2 a b$$

Short answer

The simplified expression is \(29.7ab\).

Step-by-step solution

Combine Like Terms

Since both terms, \(8.5ab\) and \(-21.2ab\), have the same variables with the same exponents, they are like terms and can be combined by adding their coefficients.

Subtract the Coefficients

To combine the like terms, subtract the coefficient of the second term from the coefficient of the first term: \(8.5 - (-21.2) = 8.5 + 21.2\).

Simplify the Expression

Add the two coefficients to get the simplified result: \(8.5 + 21.2 = 29.7\). Thus, the simplified expression is \(29.7ab\).

Key concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. Think of them as the building blocks of algebra. Variables, which are symbols like a, b, or x, can represent unknown values or values that can change. In our exercise, 8.5ab and -21.2ab are algebraic expressions where a and b are variables, and 8.5 and -21.2 are coefficients representing numbers multiplied by the variables.

Understanding algebraic expressions is crucial, as they are used to model real-world situations and solve problems in various fields, like science and engineering. Remember, expressions can become quite complex, but the principles for dealing with them remain consistent. We combine like terms or simplify them down to make these expressions more manageable and understandable.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form, making them easier to work with. It’s like tidying up a messy room so that it’s easier to move around. When we talk about 'combining like terms,' we’re essentially looking for terms in the expression that are the same, or 'like' each other, and combining them into a single term. This is a common method for simplifying expressions.

To identify like terms, look for terms that have the exact same variable parts, which means they have the same variables raised to the same powers. For instance, 8.5ab and -21.2ab are like terms because both have the variables a and b raised to the same power, which is 1 in both cases. By adding their coefficients, we combine the terms to simplify the expression.

Arithmetic Operations

Arithmetic operations are the foundation of mathematics, including addition, subtraction, multiplication, and division. These operations help us manipulate numbers and solve equations. In our exercise, we use subtraction and addition to combine like terms. We first took the opposite of subtraction, which is adding the opposite number, turning 8.5 - (-21.2) into 8.5 + 21.2.

Arithmetic isn't just about plugging and chugging numbers; it requires understanding the relationship between numbers and operations. Mastering these operations is vital, as they are not only used in algebra but across all areas of math and beyond, into everyday life scenarios like budgeting, cooking measurements, or calculating time.