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: The base-ten system writes any number with ten digits where each place is ten times the place to its right.

Common stuck point: The procedure for base-ten system is the easy part; the trap is thinking places grow by +10 instead of x10. Asking "Is each place worth exactly ten times the place to its right?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is each place worth exactly ten times the place to its right?

Worked Examples

Example 1

easy
Express 5,3045{,}304 as a sum of powers of 10.

Answer

5×103+3×102+4×1005 \times 10^3 + 3 \times 10^2 + 4 \times 10^0

First step

1
Identify each digit: 55 (thousands), 33 (hundreds), 00 (tens), 44 (ones).

Full solution

  1. 2
    Write each as a power of 10: 5×103+3×102+0×101+4×1005 \times 10^3 + 3 \times 10^2 + 0 \times 10^1 + 4 \times 10^0.
  2. 3
    Verify: 5000+300+0+4=5,3045000 + 300 + 0 + 4 = 5{,}304.
The base-ten system assigns each position a power of 10: ones (10010^0), tens (10110^1), hundreds (10210^2), thousands (10310^3), 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.

Example 4

medium
Write 52,60452{,}604 in expanded form using powers of 1010.

Example 5

medium
Write the number that has 44 ten-thousands, 00 thousands, 77 hundreds, 22 tens, and 99 ones.

Example 6

medium
Decompose 1,2351{,}235 as a sum involving place-value parts.

Example 7

hard
Why is multiplying by 10001000 the same as shifting digits three places left?

Example 8

hard
Compare 3×1043 \times 10^4 to 30×10330 \times 10^3. Are they equal?

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 10010^0, 10110^1, 10210^2, and 10310^3?

Example 2

medium
A number N=3×104+7×102+5×100N = 3 \times 10^4 + 7 \times 10^2 + 5 \times 10^0. What is NN?

Example 3

easy
What is the value of the digit 3 in the number 35?

Example 4

easy
What is the value of the digit 7 in 472?

Example 5

easy
Write 4 hundreds, 0 tens, 7 ones as a number.

Example 6

easy
What is 10010^0?

Example 7

easy
Which place is worth ten times the tens place?

Example 8

easy
Expand 256 into hundreds, tens, and ones.

Example 9

easy
What is the place value of the rightmost digit in any whole number?

Example 10

easy
How many ones make one ten?

Example 11

medium
In 3,482, what is the value of the digit 4?

Example 12

medium
Write 'six thousand forty' as a numeral.

Example 13

medium
How many times larger is the digit 5's value in 500 than in 50?

Example 14

medium
Express 705 in expanded form using powers of ten.

Example 15

medium
What number is 10 times 47?

Example 16

medium
In 8,888, is the value of the leftmost 8 ten times or a thousand times the rightmost 8?

Example 17

medium
Round 4,762 to the nearest hundred using place value.

Example 18

medium
What digit must go in the blank so 2_6 is closest to 250: 2, 4, or 5?

Example 19

challenge
A 3-digit number has digits that are consecutive increasing integers and sums to 12. What is the number?

Example 20

challenge
Convert the number 13 into base ten 'tens and ones' and explain why we cannot write a single digit '13' in one place.

Example 21

challenge
In a number, the hundreds digit is twice the ones digit, the tens digit is 0, and the ones digit is 4. What is the number?

Example 22

medium
In 1,940, what is the value of the digit 9?

Example 23

easy
What is the value of the digit 55 in 5,2685{,}268?

Example 24

easy
Write the number for 66 thousands, 33 hundreds, 00 tens, and 99 ones.

Example 25

easy
How many tens are in 720720?

Example 26

easy
Write the expanded form of 608608 using place-value labels.

Example 27

easy
Which is bigger, 987987 or 1,0021{,}002?

Example 28

medium
What is 46×10046 \times 100?

Example 29

medium
What is 8,300÷1008{,}300 \div 100?

Example 30

medium
How many times bigger is the value of the 55 in 5,0075{,}007 than the 55 in 5050?

Example 31

medium
Round 3,4823{,}482 to the nearest hundred.

Example 32

medium
In 7,5307{,}530, what is the difference between the values of the 77 and the 55?

Example 33

medium
Write the smallest 4-digit number using digits 4,0,7,14, 0, 7, 1 exactly once.

Example 34

hard
Write the largest 4-digit number using digits 4,0,7,14, 0, 7, 1 exactly once.

Example 35

hard
What is the value of the digit 99 in 9,087,2139{,}087{,}213?

Example 36

hard
How many zeros does 10710^7 have when written in standard form?

Example 37

hard
What number is 1,0001{,}000 more than 48,27348{,}273?

Example 38

hard
What digit goes in the tens place of 3×103+5×101+23 \times 10^3 + 5 \times 10^1 + 2?

Example 39

challenge
A number's digits sum to 99. The hundreds digit is twice the tens digit, and the ones digit equals the hundreds digit. What is the smallest such 3-digit number?
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Background Knowledge

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

place value