Problem 25 - Elementary and Intermediate Algebra

Question

Use the commutative law of multiplication to write an equivalent expression. $$ 5(a+1) $$

Step-by-Step Solution

Answer

(a+1)5

Step 1 - Understand the Commutative Law of Multiplication

The commutative law of multiplication states that the order in which two numbers are multiplied does not change the product: \[ a \times b = b \times a \].

Step 2 - Identify the Multiplication in the Expression

In the expression \( 5(a+1) \), the multiplication is between the constant 5 and the binomial \(a+1\).

Step 3 - Apply the Commutative Law

By applying the commutative law, switch the order of the multiplication: \[ 5(a+1) = (a+1)5 \].

Step 4 - Write the Final Equivalent Expression

The expression \( 5(a+1) \) is equivalent to \( (a+1)5 \).

Key Concepts

Equivalent Expressions

Equivalent expressions are expressions that look different but have the same value. For example, you might write a math problem in two different ways, but they can still give you the same answer. In the given exercise, \(5(a+1)\) and \((a+1)5\) are equivalent expressions. Even though we changed the order of multiplication, the value remains the same. This trick can often simplify solving algebra problems. Swap terms around to find the simplest form.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations (like addition, subtraction, multiplication, and division). For example, in the exercise \(5(a+1)\), we have:

• Numbers: 5
• Variable: a
• Operation: Addition and Multiplication

Algebraic expressions can be simple like \(x+3\) or complex like \((2a+b)^2-5c\). These forms help solve for unknown values.

Distributive Property

The distributive property is a handy rule in algebra that allows us to multiply a sum by multiplying each addend separately and then adding the products. It expresses that \ a(b+c)=ab+ac \. If the original exercise was \(5(a+2)\), we would use the distributive property to write it as \[ 5a + 5 \]. This method makes it easy to deal with parentheses and simplify equations. Remember, you break the sum or difference inside the parentheses into smaller parts and then multiply those parts individually.