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Special Powers: Squares and Cubes

Some exponential forms get their special names from the square and cube of geometry.
2
3
x is read as x squared and x is read as x cubed.
2
For example: 3 is read as 3 squared or 3 raised to the power 2.
And 7 is read as 7 cubed or 7 raised to the power 3.
3
3
2
In 3 , 3 is the base and 2 is the exponent whereas in 7 , 7 is the base and 3 is the exponent.
Now, let us take any integer a as the base, and write the numbers as,
2
a × a = a (read as ‘a squared’ or ‘a raised to the power 2’)
3
a × a × a = a (read as ‘a cubed’ or ‘a raised to the power 3’)
4
a × a × a × a = a (read as ‘a raised to the power 4’ or ‘4th power of a’)
7
a × a × a × a × a × a × a = a (read as ‘a raised to the power 7’ or ‘7th power of a’)
and so on.

In general, if a is any integer and n is a natural number, then

a = a × a × a × a × . . . n times
n
Exponential Expanded or
form Product form
n
In a , a is the base and n is the exponent or power. a is read as ‘a to the power n’ or ‘a raised
n
to the power n’.
3 2
Also, a × a × a × b × b can be expressed as a b (read as a cubed b squared).
2 4
a × a × b × b × b × b can be expressed as a b (read as a squared b raised to the power 4).

create and solve
Give an example of each of the following conditions:
• A number whose base is 10 and exponent is an even number.
• A number with a negative integer base and exponent is 3.
• A 5-digit number is written in expanded form in terms of a power of 10 .
• The square and cube of an odd number.
Write some more such conditions in your notebook and ask your friend to give examples from them.

Example 1: Write the base and the exponent in the following notation:

(a) 4 5 (b) (−3) 7 (c) 5 0 (d) 10 5
Solution: (a) In 4 , base = 4, and exponent = 5
5
7
(b) In (−3) , base = −3, and exponent = 7
0
(c) In 5 , base = 5, and exponent = 0
5
(d) In 10 , base = 10, and exponent = 5
Example 2: Write the following in exponential form.

(a) 32, whose base is 2 (b) 27, whose base is 3 (c) (–125), whose base is (–5)
5
Solution: (a) 32 = 2 × 2 × 2 × 2 × 2 = 2 (b) 27 = 3 × 3 × 3 = 3 3
(c) (–125) = (−5) × (−5) × (−5) = (−5) 3

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