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Chapter 1: Problem 7

Perform each computation. Make use of appropriate rules to simplify each problem. $$ 45(200) $$

Short Answer

Expert verified
9000

Step by step solution

01

Understand the Problem

We are asked to multiply two numbers. The numbers given are 45 and 200.
02

Use the Distributive Property

Rewrite the multiplication problem using the distributive property. Break down the numbers into simpler components: Use the fact that 200 can be written as 2 脳 100.So, we have: 45 脳 200 = 45 脳 (2 脳 100).
03

Rearrange Terms

According to the associative property of multiplication, we can group the terms differently without changing the result:45 脳 (2 脳 100) = (45 脳 2) 脳 100.
04

Multiply the Simpler Terms

First, multiply the simpler terms inside the parentheses: 45 脳 2 = 90.
05

Final Multiplication Step

Now multiply the result by 100:90 脳 100 = 9000.

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Key Concepts

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

distributive property
The distributive property is one of the most important properties in math. It allows you to break down complex multiplication into simpler parts. This is particularly useful when dealing with large numbers. The distributive property states that for any numbers a, b, and c: \[ a(b + c) = ab + ac \]. In our example, we break down 200 into two simpler numbers: 2 and 100. So 45 脳 200 becomes 45 脳 (2 脳 100). By distributing 45 over 2 and 100, you can make the problem easier to handle.
associative property
The associative property of multiplication tells us that it doesn't matter how we group the numbers when multiplying. You can rearrange the parentheses in any way, and the result will be the same. The associative property is defined as: \[ (a 脳 b) 脳 c = a 脳 (b 脳 c) \]. This property is used in our problem when we rearrange the terms: 45 脳 (2 脳 100) becomes (45 脳 2) 脳 100. By grouping 45 and 2 first, you can simplify the calculation to eventually get 9000.
multiplication of numbers
Multiplication is one of the basic arithmetic operations. It helps to scale one number by another. The numbers you multiply are called factors, and the result is called the product. In our example, 45 and 200 are the factors, and 9000 is the product. Using properties like distributive and associative makes multiplication easier, especially for large numbers. Always break down the problem into manageable parts and then multiply step by step to get the final product.

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