Problem 28 - Elementary Algebra

Question

Use the commutative property of addition and multiplication to write expressions for an equal number for the following problems. You need not perform any calculations. $$7(2+b)$$

Step-by-Step Solution

Answer

Question: Use the commutative property of addition and multiplication to rewrite the expression: $$7(2+b)$$ Answer: $$(b+2)7$$

Step 1: Recall commutative property

Let's recall the commutative property: For addition: $$a+b=b+a$$ For multiplication: $$ab=ba$$ Now we will use these properties to rewrite the given expression.

Step 2: Apply commutative property to the expression

Given expression: $$7(2+b)$$ First, we apply commutative property to the addition inside the brackets: $$7(2+b) = 7(b+2)$$ Next, apply commutative property to the multiplication: $$7(b+2) = (b+2)7$$ So, the final expression by using commutative properties is: $$(b+2)7$$

Key Concepts

Properties of Addition

When it comes to understanding how numbers interact during addition, the commutative property plays an essential role. This property simply indicates that no matter in which order we add numbers, the result will always be the same. Mathematically, this is expressed as \( a + b = b + a \). This means that if we have two numbers, such as 4 and 5, adding them up in any order, whether it's \( 4 + 5 \) or \( 5 + 4 \) will yield the same sum, which is 9 in this case.

Applying this to our original problem, where the expression includes \( 7(2 + b) \), we can rearrange the terms within the parentheses without altering the value of the expression. This versatility can be powerful for simplifying algebraic equations and is a fundamental aspect of elementary algebra.

Properties of Multiplication

Similar to addition, multiplication has its own commutative property. This property states that the order in which we multiply numbers does not affect the product. Formally, the rule is written as \( ab = ba \), which indicates that the multiplication of 'a' and 'b' yields the same product as 'b' and 'a'. For example, \( 3 \times 4 = 4 \times 3 \) both give us 12.

In the context of the problem \( 7(2 + b) \), once we've utilized the commutative property for addition to get \( 7(b + 2) \), we can further use the commutative property of multiplication to flip the expression into \( (b + 2)7 \), reassuring us that this alteration keeps the expression's value unchanged.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. These expressions are the backbone of algebra and can be as simple as \( b + 2 \) or as complex as \( 4x^2 - 3xy + 7 \). When we are given an expression such as \( 7(2 + b) \), it implies a multiplication of 7 by the sum of 2 and 'b'. In our example, we leveraged the previously mentioned commutative properties to rewrite the expression, which is a standard technique in algebra for simplifying and manipulating expressions.

Elementary Algebra

Elementary algebra is the branch of mathematics that deals with solving equations and understanding how various algebraic concepts interact. Fundamental principles like the commutative property are part of what makes algebra both interesting and useful in various applications. Grasping these concepts allows for the solving of more complicated problems in future mathematical pursuits.

The properties we have discussed are critical tools for working efficiently with algebraic expressions. By understanding and applying these properties, students can unlock a deeper comprehension of algebra and improve their problem-solving strategies.