Back to Unit 1 numeration

Notes on Exponents and Place Value

 An exponent is used to tell how many times a number, called the base is used as a factor:  It is written and a small, raised number to the right of its base:

For 54  5 is the base and 4 is the exponent.  This means multiply the 5 by itself, four times.

5 X 5 X 5 X 5    5 is used as a factor four times.

Sometimes we refer to exponentiation as raising a number to a power.  For example 34 can be read as three to the fourth power.

Any number raised to the 2nd power is said to be squared and any number raised to the third power is said to be cubed.


Standard form is 625

Expanded form is       5 X 5 X 5 X 5                    The easiest way of doing this is to multiply every pair:

                                  25   X   25 =  625

Power of Ten:  Any number that is an exponential form of 10.  For example 100 is 102, 10,000 is 104, 10 is 101.

 How do we write the expanded form of numbers using exponents for its power of ten:  We sum up the value of each digit by multiplying by its power of ten in exponential form.

We write the number 148,582 using different formats:

Standard form: 148,582

Expanded form: (1 X 100,000) + (4 X 10,000) + (8 X 1,000) + (5 X 100) + (8 X 10) + (2 X 1)

Expanded form using exponents: (1 X 105) + (4 X 104) + (8 X 103) + (5 X 102) + (8 X 101) + (2 X 100)

Any number raised to the power of 1 always equals that number:

    example: 101 = 10 and 41 = 4

Any number raised to the zero power always equals one:

    example : 100 = 1 and 40 = 1