Question

Remove parentheses and simplify. $$2(a+3 b)$$

Short answer

2a + 6b

Step-by-step solution

Apply the Distributive Property

Use the distributive property a(b + c) = ab + ac to remove the parentheses. Multiply each term inside the parentheses by the factor outside the parentheses.

Multiply Each Term Inside the Parentheses by 2

Multiply 2 by a to get 2a. Multiply 2 by 3b to get 6b.

Combine the Results

Combine the products from the previous step to get the simplified expression without parentheses.

Key concepts

Understanding the Distributive Property

When simplifying algebraic expressions, the distributive property is a fundamental tool. It allows you to remove parentheses by distributing a multiplier to each term within the parentheses. For instance, when we have an expression like \(2(a+3b)\), we want to apply the distributive property, which can be expressed mathematically as \(a(b+c) = ab+ac\).

So, if we apply it to our example, we multiply \(2\) by \(a\) and \(3b\) separately: \[2 \times a = 2a\] and \[2 \times 3b = 6b\]. This property is essential because it helps us transform and simplify expressions so that we can solve them more easily or combine them with other terms.

Combining Like Terms

After using the distributive property, the next step in simplifying expressions is to combine like terms. Like terms are terms within an expression that have the same variable part and are raised to the same power. For example, \(2a\) and \(3a\) are like terms, but \(2a\) and \(3b\) are not.

In the expression we obtained from the distributive property, \(2a+6b\), there are no like terms to combine because the terms have different variables. However, if our expression was \(2a+3a\), we could combine these to get \(5a\), simplifying our expression further. It's like combining apples with apples; you wouldn't add apples to oranges. Always look for and combine like terms to make your algebraic expressions as simple as possible.

Algebraic Expressions Essentials

An algebraic expression is a collection of numbers, variables, and operations. In our example of \(2a+6b\), \(2a\) and \(6b\) are terms that contribute to the overall expression.

An important aspect of interpreting these expressions is identifying each part: the coefficients, which are the numerical part of the terms (in our case, \(2\) and \(6\)), and the variables (here, \(a\) and \(b\)). The beauty of algebraic expressions is that they allow us to perform calculations with unknown values, which we can later determine if we have enough information. Mastering how to work with algebraic expressions can open doors to understanding more complex mathematics and solving a variety of real-world problems.