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Chapter 1: Problem 36

Write each of the following numbers in expanded form. $$20,088,060$$

Short Answer

Expert verified
The expanded form of 20,088,060 is \(2 \times 10,000,000 + 8 \times 10,000 + 8 \times 1,000 + 6 \times 10\).

Step by step solution

01

Identify Each Place Value

Start by understanding the place value of each digit in the given number \(20,088,060\). From left to right, the digits represent the ten millions, millions, hundred thousands, ten thousands, thousands, hundreds, tens, and units: again here, the number is 20,088,060.
02

Break Down Each Digit to its Place Value

Based on the identified place values, split the number \(20,088,060\) according to each digit's value:- Tens of millions = \(2 \times 10,000,000\)- Millions = \(0\)- Hundred thousands = \(0\)- Ten thousands = \(8 \times 10,000\)- Thousands = \(8 \times 1,000\)- Hundreds = \(0\)- Tens = \(6 \times 10\)- Units = \(0\)
03

Write Number in Expanded Form

Now, express the number \(20,088,060\) by writing out the value of each digit:\[2 \times 10,000,000 + 0 \times 1,000,000 + 0 \times 100,000 + 8 \times 10,000 + 8 \times 1,000 + 0 \times 100 + 6 \times 10 + 0\].

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Key Concepts

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

Place Value
A fundamental concept in understanding expanded form is place value. When you look at a number, each digit has its own place, and this place determines its value. Imagine it like rows of seats in a theater, where each row represents a different power of 10.
  • Tens of Millions: This is the left-most position in our example number, where the digit is multiplied by 10,000,000.
  • Millions, Thousands, Hundreds: These places come next, with each one a power of 10 times smaller.
  • Units: The right-most place, where the digit is simply how many ones you have.
When you "read" a number, understanding each digit's place value helps you say exactly how much that part of the number is worth. This is key to writing numbers in different forms, like expanded form.
Prealgebra
Working with numbers in expanded form is a fantastic exercise in prealgebra. Prealgebra lays the foundation for algebra and involves mainly understanding how numbers work, and an essential part of this is manipulating numbers into different forms, including expanded form.
  • By practicing breaking a number into its component place values, students develop an intuition for how numbers are constructed, leading to a better grasp of mathematical operations like addition and multiplication.
  • This practice also aids in learning how to solve equations, as expanded form represents a number in a form similar to how one might set up parts of an algebraic equation.
  • It's all about simplifying complex problems, allowing you to look at a number piece by piece, rather than all at once.
Thus, mastering expanded form in mathematics strengthens basic arithmetic skills necessary for tackling algebraic problems later on.
Number Representation
There are various ways to represent numbers, and one important method is expanded form. While the standard form of a number helps with everyday counting, expanded form is invaluable for understanding the number's structure in detail.
  • Expanded Form: Like in our example of 20,088,060, expanded form shows the value of each digit according to its place value, expressing the number as a sum of each digit multiplied by its place value.
  • Why Use It? Using expanded form helps when learning addition or subtraction of large numbers, offering a clear view of each component.
  • Comparison and Estimation: It's easier to compare magnitudes of numbers or estimate sums and differences when each place value is clear.
Overall, expanded form is a powerful tool in number representation, enhancing both basic and complex arithmetic understanding.

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