Base-Ten System Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Base-Ten System.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The positional numeral system using ten as its base, where each digit's value depends on its position, with each place worth ten times the place to its right.

We group things by tens—probably because we have 10 fingers.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Position determines value through powers of 10: \ldots 1000, 100, 10, 1, 0.1, 0.01 \ldots

Common stuck point: Not seeing that other bases (binary, hexadecimal) work the same way.

Sense of Study hint: Try bundling objects into groups of ten, then groups of ten-tens (hundreds), to physically see how the system works.

Worked Examples

Example 1

easy
Express 5{,}304 as a sum of powers of 10.

Solution

  1. 1
    Identify each digit: 5 (thousands), 3 (hundreds), 0 (tens), 4 (ones).
  2. 2
    Write each as a power of 10: 5 \times 10^3 + 3 \times 10^2 + 0 \times 10^1 + 4 \times 10^0.
  3. 3
    Verify: 5000 + 300 + 0 + 4 = 5{,}304.

Answer

5 \times 10^3 + 3 \times 10^2 + 4 \times 10^0
The base-ten system assigns each position a power of 10: ones (10^0), tens (10^1), hundreds (10^2), thousands (10^3), etc. Writing a number in this form makes its structure explicit and connects place value to exponents.

Example 2

medium
Why does multiplying any whole number by 10 append a zero? Explain using the base-ten structure.

Example 3

medium
Write 3,407 in expanded form using powers of 10.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
What is the value of 10^0, 10^1, 10^2, and 10^3?

Example 2

medium
A number N = 3 \times 10^4 + 7 \times 10^2 + 5 \times 10^0. What is N?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

place value