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Chapter 2: Problem 107

Apply the distributive property to each expression and then simplify the result. $$2(3+4)$$

Short Answer

Expert verified
The expression simplifies to 14.

Step by step solution

01

Apply the Distributive Property

The distributive property states that for any numbers \(a\), \(b\), and \(c\), \(a(b+c) = ab + ac\). In this problem, we apply it by distributing the \(2\) to both \(3\) and \(4\), which gives us \(2 \times 3 + 2 \times 4\).
02

Perform the Multiplications

Now, we carry out the multiplications: \(2 \times 3 = 6\) and \(2 \times 4 = 8\). The expression now becomes \(6 + 8\).
03

Simplify the Expression

Finally, add the numbers obtained from the multiplication: \(6 + 8 = 14\). So the expression simplifies to \(14\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prealgebra
Prealgebra forms the core foundation for all higher-level math concepts. It involves basic arithmetic, number sense, and introduces algebraic thinking in a simple way to help students transition smoothly to more advanced topics. In prealgebra, students focus on:
  • Understanding numbers and how they relate to one another.
  • Learning operations such as addition, subtraction, multiplication, and division.
  • Beginning to think about mathematical problems more abstractly.
Prealgebra serves as a bridge, ensuring students are comfortable with these fundamental operations. This solid understanding enables them to tackle more complex problems involving variables and unknowns when they move on to algebra.
Mathematical Expressions
Mathematical expressions are combinations of numbers, variables, and operators that represent a particular value or set of values. They do not have an "equals" sign because they are not equations. Expressions can look like simple numbers or more complex strings using operations such as addition, subtraction, multiplication, and division.
  • An example of a mathematical expression is \(2(3+4)\).
  • Expressions can be simplified using properties such as the distributive property, which helps break down expressions into easier parts.
When you simplify or work with expressions, your goal is often to reduce them to their simplest form, which makes it easier to spot patterns or understand their meanings. Learning how to manipulate and simplify mathematical expressions is a key skill in all areas of math and science.
Simplification
Simplification makes expressions easier to understand and work with by reducing them to a simpler form. In mathematics, especially in algebra, you simplify expressions by:
  • Applying properties like the distributive property, which allows you to multiply a number by a group of numbers added together, e.g., \(a(b+c) = ab + ac\).
  • Performing operations, such as combining like terms or carrying out arithmetic calculations.
  • Breaking down complex expressions into simpler ones.
In our example, \(2(3+4)\), applying simplification involved:1. Using the distributive property to transform the expression into \(2 \times 3 + 2 \times 4\).2. Carrying out the multiplication to get \(6+8\).3. Finally, adding the results to achieve a simplified expression, which is \(14\).Simplification helps make mathematical solutions more concise and reveals the core concepts easily, which is especially beneficial when dealing with extensive and intricate algebraic expressions.

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Most popular questions from this chapter

Draw a number line from 10 to 10 and use it to add the following numbers. $$-2+(-7)$$

Without pencil and paper or a calculator. Is \(-20\) closer to \(-40\) or \(10 ?\)

Add or subtract. $$276+32+4,005$$

Use any of the rules developed in this chapter and the rule for order of operations to simplify each of the following expressions as much as possible. [Examples 6鈥�9] $$\frac{4(-7)}{-28}$$

Subtract. $$-7-(-4)$$
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