Question

Remove parentheses and simplify. $$-(a+3.92)-(a-4.14)$$

Short answer

-2a - 0.22

Step-by-step solution

Distribute the negative sign

Apply the negative sign to each term within both sets of parentheses. This is done by changing the sign of each term inside the parentheses. Recall that subtracting a number is the same as adding its negative.

Combine like terms

After distributing the negative signs, combine like terms. This means adding or subtracting terms that contain the same variable or are constants.

Simplify the expression

Perform the arithmetic operation to simplify the expression to its simplest form. Here, that means just combining the constants.

Key concepts

Distributive Property

Understanding the distributive property is vital when you're trying to remove parentheses and simplify algebraic expressions. It's a tool that allows you to multiply a single term by each of the terms inside a parenthesis. The mathematical representation of the distributive property is expressed as: \[a(b+c) = ab + ac\]. In this formula, \(a\) is distributed to both \(b\) and \(c\).

When you encounter a negative sign before the parentheses, like in our exercise, it's equivalent to multiplying by -1. Therefore, every term inside the parenthesis changes its sign when the negative is distributed across. This initial step is crucial because it sets up the expression for further simplification.

Combining Like Terms

Once the distributive property has been applied, the next step in the process of simplification is combining like terms. Like terms are terms that have the same variables raised to the same power. In the context of our exercise, there are two types of like terms: the variable \(a\) and the numerical constants.

To successfully combine like terms, simply add or subtract the coefficients (the numbers in front of the variables) and keep the variable part unchanged. For constants, just add or subtract the numbers as you normally would in arithmetic. For instance, if you have \(2a\) and \(3a\), you can combine them to get \(5a\); they are like terms because they both have the variable \(a\).

Arithmetic Operations

The final stretch in simplifying algebraic expressions involves basic arithmetic operations: addition, subtraction, multiplication, and division. Once like terms are combined, you're often left with a more straightforward expression that requires you to perform one or more of these operations.

In our exercise, after combining like terms, what remains may be constants that need to be added or subtracted. These operations are the ones you've learned since elementary school, and they are the last steps to arriving at the simplest form of an expression. It's like putting the final touches on a painting – the details may be small, but they're crucial to the final product.