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

A number system using ten symbols (0-9) where each place represents a power of ten.

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.

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