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Chapter 1: Problem 16

Write the product of 23 and \(14,\) using three different notations. (Answers may vary.)

Short Answer

Expert verified
The product of 23 and 14 is 322.

Step by step solution

01

- Standard Multiplication

First, compute the product of 23 and 14 using standard multiplication. 23 脳 14 鈥斺�斺�斺�斺�斺�斺�� 92 (23 脳 4) 230 (23 脳 10, shift one position to the left) 鈥斺�斺�斺�斺�斺�斺�� 322 (Adding both products together) The standard multiplication yields 322.
02

- Using Algebraic Distribution

Express 23 and 14 in algebraic form: 23 = 20 + 3 14 = 10 + 4 Using distribution: (20 + 3)(10 + 4) 20 脳 10 + 20 脳 4 + 3 脳 10 + 3 脳 4 200 + 80 + 30 + 12 Adding all terms: 200 + 80 + 30 + 12 = 322 The product using algebraic distribution is 322.
03

- Using Calculators

Another approach is to simply use a calculator. Enter the values: 23 脳 14 = This yields the same result of 322. Hence, the product, using a calculator, is 322.

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Key Concepts

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

Standard Multiplication
Standard multiplication is a traditional method often taught in schools. It's a step-by-step way of multiplying two numbers. For example, to find the product of 23 and 14, follow these steps:
  • First, multiply 23 by 4 (the ones place of 14). This gives you 92.
  • Next, multiply 23 by 10 (the tens place of 14). Remember to shift this one place to the left, giving you 230.
  • Finally, sum both results: 92 + 230. This gives the final product, 322.
Notice how we break down the problem into smaller, manageable parts. Each multiplication step stays straightforward, reducing errors and simplifying the process.
Algebraic Distribution
Algebraic distribution involves breaking down numbers into simpler parts. This is also known as the distributive property of multiplication. To illustrate, consider 23 as 20 + 3 and 14 as 10 + 4. Now, distribute each part with the others:
  • First, multiply 20 by 10 to get 200.
  • Next, multiply 20 by 4 to get 80.
  • Then, multiply 3 by 10 to get 30.
  • Finally, multiply 3 by 4 to get 12.
Add these four results together (200 + 80 + 30 + 12) to get 322. By breaking the problem into smaller sections, it becomes easier to handle. This method is particularly helpful in more complex multiplications.
Calculator Usage
Using a calculator simplifies multiplication, especially for larger numbers or quick calculations. Ensure you carefully enter the numbers into the calculator. For example, input 23, then press the multiplication sign (脳), followed by 14. Press 'equals' (=), and you'll get the result: 322.
Be cautious when using calculators to avoid errors from wrong button presses or incorrect entries. Double-check your numbers and use clear functions to maintain accuracy. Calculators are excellent for confirming results obtained through manual methods like standard multiplication or algebraic distribution.

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