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Chapter 1: Problem 29

Write each of the following numbers in expanded form. $$3,462,577$$

Short Answer

Expert verified
The expanded form of 3,462,577 is \(3,000,000 + 400,000 + 60,000 + 2,000 + 500 + 70 + 7\).

Step by step solution

01

Understanding Expanded Form

Writing a number in expanded form means expressing it as the sum of its digits multiplied by their respective place values.
02

Identify Place Values

Break down the number according to its place values: 3 in the millions place, 4 in the hundred thousands place, 6 in the ten thousands place, 2 in the thousands place, 5 in the hundreds place, 7 in the tens place, and 7 in the ones place.
03

Multiply Digits by Place Values

For each digit, multiply it by its place value: \(3 \times 1,000,000\), \(4 \times 100,000\), \(6 \times 10,000\), \(2 \times 1,000\), \(5 \times 100\), \(7 \times 10\), and \(7 \times 1\).
04

Write the Expanded Form

Combine these products into an expression: \(3,000,000 + 400,000 + 60,000 + 2,000 + 500 + 70 + 7\).

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Key Concepts

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

Understanding Place Values
When dealing with numbers, place values play a crucial role in determining the value of each digit. Place values are like addresses, showing where each digit sits in a number and what it stands for. In our given number, 3,462,577, each digit has its unique place value.

Here's how it works:
  • Millions: The first digit, 3, is in the millions place, signifying three million.
  • Hundred Thousands: The digit 4 stands for four hundred thousand.
  • Ten Thousands: The 6 represents sixty thousand.
  • Thousands: The 2 indicates two thousand.
  • Hundreds: The 5 means five hundred.
  • Tens: The first 7 is seventy.
  • Ones: The last 7 indicates seven.
These place values help us see the structure of the number and set the stage for expanding it.
The Role of Multiplication in Expanded Form
Multiplication is a key operation when writing numbers in expanded form. It allows you to convert each digit into its full numerical value by multiplying it with its place value. This helps you understand precisely what each digit contributes to the total number.

For the number 3,462,577, you multiply each digit by its place value:
  • The 3 is multiplied by 1,000,000, since it's in the millions place, becoming 3,000,000.
  • The 4 is multiplied by 100,000, as it's in the hundred thousands place, making 400,000.
  • The 6 is multiplied by 10,000, giving us 60,000.
  • The 2 is multiplied by 1,000, resulting in 2,000.
  • The 5 is multiplied by 100, giving 500.
  • The first 7 is multiplied by 10, resulting in 70.
  • The last 7 is multiplied by 1, giving a value of 7.
This multiplication is what expands each digit into its full description, allowing you to describe the combination of these expanded values in the next step.
Combining with Addition
After using multiplication to express each digit's full value, the final step is to use addition to combine them all back together. This step brings the expanded parts of a number into a complete and clear expression of the entire value.

Look at what we've built:
  • The 3 was expanded to 3,000,000.
  • The 4 became 400,000.
  • The 6 turned into 60,000.
  • The 2 expanded to 2,000.
  • The 5 became 500.
  • The first 7 expanded to 70.
  • The last 7 became 7.
Adding all these expanded components: \[3,000,000 + 400,000 + 60,000 + 2,000 + 500 + 70 + 7\]. This expression clearly illustrates how each part of the number contributes to the whole. It鈥檚 about connecting each separate piece through addition to understand the number's overall composition.

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Most popular questions from this chapter

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