250 MATHEMATICS
We have used numbers like 10, 100, 1000 etc., while writing numbers in an expanded
form. For example, 47561 = 4 × 10000 + 7 × 1000 + 5 × 100 + 6 × 10 + 1
This can be written as 4 × 10^4 + 7 ×10^3 + 5 × 10^2 + 6 × 10 + 1.
Try writing these numbers in the same way 172, 5642, 6374.
In all the above given examples, we have seen numbers whose base is 10. However
the base can be any other number also. For example:
81 = 3 × 3 × 3 × 3 can be written as 81 = 3^4 , here 3 is the base and 4 is the exponent.
Some powers have special names. For example,
102 , which is 10 raised to the power 2, also read as ‘10 squared’ and
103 , which is 10 raised to the power 3, also read as ‘10 cubed’.
Can you tell what 5^3 (5 cubed) means?
53 means 5 is to be multiplied by itself three times, i.e., 5^3 = 5 × 5 × 5 = 125
So, we can say 125 is the third power of 5.
What is the exponent and the base in 5^3?
Similarly, 2^5 = 2 × 2 × 2 × 2 × 2 = 32, which is the fifth power of 2.
In 2^5 , 2 is the base and 5 is the exponent.
In the same way, 243 = 3 × 3 × 3 × 3 × 3 = 3^5
64 = 2 × 2 × 2 × 2 × 2 × 2 = 2^6
625 = 5 × 5 × 5 × 5 = 5^4Find five more such examples, where a number is expressed in exponential form.
Also identify the base and the exponent in each case.You can also extend this way of writing when the base is a negative integer.
What does (–2)^3 mean?
It is (–2)^3 = (–2) × (–2) × (–2) = – 8
Is (–2)^4 = 16? Check it.
Instead of taking a fixed number let us take any integer a as the base, and write the
numbers as,
a × a =a^2 (read as ‘a squared’ or ‘a raised to the power 2’)
a × a × a =a^3 (read as ‘a cubed’ or ‘a raised to the power 3’)
a × a × a × a =a^4 (read as a raised to the power 4 or the 4th power of a)
..............................
a × a × a × a × a × a × a = a^7 (read as a raised to the power 7 or the 7th power of a)
and so on.
a × a × a × b × b can be expressed as a^3 b^2 (read as a cubed bsquared)TRY THESE