We use indices to simplify multiplication problems that use the same number more than once.
When the index is a natural number, the index indicates how many times to multiply the number by itself.
By the end of this section you should:
● understand that indices (powers) represent repeated multiplication
● be able to convert between expanded and index form
● be able to fi nd the value of numbers expressed in index form
Consider the calculation 2 × 2 × 2 × 2 × 2 We are simply multiplying 2 by itself 5 times. A more convenient way to write this is 25
‘2 to the power of 5’. The notation 25
is index form. The 2 × 2 × 2 × 2 × 2 is known as expanded form. To convert from expanded form to index form:
1. Write the number that is repeated down fi rst. This number is called the base.
2. Count the number of times the base number appears
3. Write the number of times the base appears as a small number to the upper right of the base number. This number is called the index number. 2 × 2 × 2 × 2 × 2 = 25 Index
Appears 5 times
Base The index number can also be called the power or exponent.
The index number tells you how many times the base number should be multiplied by itself. 2
When the index number is two, the base number has been squared.
When the index number is three, the base number has been cubed.
When the index number is greater than three we say that it has been multiplied to the power of the index: 27
is read as ‘2 to the power of 7’. on a scientifi c calculator: or y x 2 2 2 × 2 = 22 = ‘two squared’
Scientifi c calculators have a ‘power’ button. This button may be labelled xy To work out 410
1. Enter 4 2. Press the index button x y 3. Enter 1
0
4. Press = You should get the answer 1 048 576.
Any number to the power of 1 will always be the number itself.
For example, 51 = 5 and 3261 = 326 or x∎ or yx 2 × 2 × 2 = 23 or x∎ . = ‘two cubed’ 2 × 2 = 4 2 2 2 × 2 × 2 = 23