Problem 256 - Prealgebra

Question

In the following exercises, multiply. (a) \(8 \times 9 \quad\) (b) \(9 \times 8\)

Step-by-Step Solution

Answer

Both \((8 \times 9)\) and \((9 \times 8)\) equal 72.

Step 1: Understand the Commutative Property of Multiplication

The commutative property of multiplication states that changing the order of factors does not change the product. Therefore, a) \(8 \times 9\) is the same as b)\( 9 \times 8 \), as both represent the same multiplication.

Step 2: Multiply the Numbers

Calculate the product of 8 and 9 using basic multiplication.a) \(8 \times 9 = 72\)b)\( 9 \times 8 = 72\).

Step 3: Verify the Result

Ensure the results are consistent. Both \((8 \times 9)\) and \((9 \times 8)\) yield the same product, which is 72.

Key Concepts

Basic Multiplication

Multiplication is one of the four basic arithmetic operations. It involves calculating the total of adding a number (called a factor) a certain number of times. For example, multiplying 8 by 9 (written as \(8 \times 9\)) involves adding 8 nine times:

• 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8.

When you perform basic multiplication, you might also use multiplication tables. These tables list the products of pairs of numbers and are useful for quick reference.

Order of Operations

In mathematics, the order in which different operations are performed is crucial. The standard rule can be remembered by PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction).However, in the case of simple multiplications like \(8 \times 9\) or \(9 \times 8\), the order of operations specifically refers to performing multiplication before addition or subtraction if they appear together.This means no matter the expression, you always carry out multiplication and division from left to right before moving on to addition and subtraction.

Verify Multiplication Results

Verification ensures the accuracy of your computations. When you multiply two numbers, you can verify your result by reversing the order, thanks to the Commutative Property of Multiplication.According to this property, \(a \times b = b \times a\). Hence, multiplying \(8 \times 9\) should yield the same product as \(9 \times 8\). Both should yield 72 when correctly multiplied.Another way to verify is to use a different method, such as breaking numbers into smaller components, also called the distributive property or using a calculator for cross-checking.Ensuring your results are accurate helps build confidence in your mathematical skills.