Question

The expression \(a[b+(c-d)]\) is equivalent to: A. \(a b+a c-a d\) B. \(a b+a c+a d\) C. \(a b+a c-d\) D. \(a b+c+d\) E. \(a b+c-d\)

Short answer

Answer: A. \(a b + a c - a d\)

Step-by-step solution

Identify the distributive property

The distributive property states that \(a(b + c) = a b + a c\). We can use this property to simplify our given expression \(a[b+(c-d)]\).

Apply the distributive property to the expression

Let's apply the distributive property to the given expression \(a[b+(c-d)]\). First, we need to distribute \(a\) to each term inside the brackets. So, we have \(a \times b + a \times (c-d)\)

Apply the distributive property again

Now we have an expression with a sum inside the brackets: \(a\times (c-d)\). We need to distribute \(a\) to each term inside the brackets, just like before. So, we have \(a \times c - a \times d\)

Rewrite the simplified expression

Now, we can rewrite the simplified expression as \(a b + a c - a d\).

Match the expression with the answer options

Our simplified expression, \(a b + a c - a d\), matches option A. So, the correct answer is: A. \(a b + a c - a d\)

Key concepts

Algebraic Expressions

One of the building blocks of algebra is the understanding of algebraic expressions. These are combinations of numbers, variables (like \(a, b, c,\) and \(d\) in our exercise), and operations (like addition and subtraction). They can represent a wide range of mathematical relationships and are key in formulating and solving equations.

An important skill when handling algebraic expressions is recognizing and applying the distributive property correctly. In our exercise, for example, the expression \(a[b+(c-d)]\) can look daunting at first, but knowing how to distribute the \(a\) to each term inside the brackets makes the problem much simpler. Understanding algebraic expressions also underpins mathematical problem solving in various fields such as engineering, economics, and physics, bridging the gap between abstract math and real-world applications.

ACT Math

Preparing for the ACT Math section involves mastering various algebraic skills, including the ability to manipulate and simplify expressions. The ACT Math test assesses a range of topics from pre-algebra to trigonometry, and strong algebraic foundation is crucial. The distributive property, as seen in our textbook problem, is a frequent test item.

For students, ACT Math practice includes working through problems that require simplifying algebraic expressions, finding equivalent expressions, and choosing the correct option from multiple choices, much like our original exercise. The key to succeeding is practice and familiarity with the types of questions asked, as well as the efficient use of test-taking strategies, such as eliminating obviously incorrect answers, which leaves fewer options to consider.

Simplifying Expressions

Simplifying expressions is a core skill in algebra. It allows us to take complex-looking expressions and rewrite them in a more manageable form. By doing so, we can more easily solve equations, compare expressions, and find solutions. The process often involves combining like terms, utilizing properties of operations, and performing arithmetic simplifications.

In our exercise, simplification involves the distributive property, which dictates how to deal with expressions involving parenthesis. We can simplify \(a[b+(c-d)]\) by multiplying \(a\) by each of the terms inside the brackets, ultimately resulting in \(a b + a c - a d\). By practicing simplifying expressions, students develop an eye for spotting shortcuts and patterns, which is especially beneficial in exams where time efficiency is essential.