In general if ‘x’ is any number and ‘n’ is any natural number, then we have . n times. The number ‘x’ is called the base and ‘n’ is called the exponent (or) the index of the exponential expression, .
Eg: In exponential form, we write 2 × 2 × 2 as read as 2 raised to the power 3.
We have, base = 2 and exponent = 3
If is any fractional number, then for any positive integer ‘m’, we have .
The reciprocal of a non zero integer ‘x’ is denoted by and defined as .
For a fractional number (where ) we have .
The reciprocal of is given by .
2. LAWS OF EXPONENTS
The product of the two powers of the same base is a power of the same base with the
index equal to the sum of the indices.
i.e., if be any rational number and m, n be positive integers, then
Power of a power. i.e., for all positive integers
Power of a product , where and ‘m’ is a positive integer.
Repeated application of this gives a more general result namely
Quotient of powers of the same base
power of a Quotient i.e., where and ‘m’ is a positive integer.
POWERS WITH ZERO AND NEGATIVE EXPONENTS
i) for every non zero real number ‘a’
ii) where m, n are positive integers and . If we denote the multiplicative inverse of an by
EXPRESSING LARGE NUMBERS IN THE STANDARD FORM
In earlier classes, we have learnt to write a number in the expanded form, as shown below:
473 = 4 ×100 + 7 × 10 + 3
3758 = 3 × 1000 + 7 × 100 + 5 × 10 + 8
30739 = 3 × 10000 + 0 × 1000 + 7 × 100 + 3 × 10 + 9
We can express these using powers of 10 in the exponential form:
In this section, we shall learn to write large numbers, using powers of 10 as shown above.
Standard Form Any number can b written as a number between 1 and 10 multiplied by a power of 10. This is called standard form of the number.
To write a number in standard form we split it into two parts multiplied together. The first part must be a number between 1 and 10 and the second, a power of 10.
For the number 56750, we start with 5.6750 in order to have a number between 1 and 10; and then we move the decimal point to the right until it is in its correct place (i.e., 56750.0).
Here , it needs to move 4 places.
The number 4 is the power of 10.