**4.1 Introduction**: An exponent is a simple notation used for **multiplying that number of identical factors**. Exponentiation is mathematical operation written in the form of a^n where n is an integer and a is any value.

a^{n} = a x a x a x……. a

a is called the base and n is called the exponent.

example: a=4; n=2

a^{n} = 4^{2} = 4 x 4 = 16

an is read as a powered n or a raised to the power n.

**4.2.1 Positive integer exponents**:

n= positive integer

a^{2} = a x a is called the square of a

a^{3} = a x a x a is called the cube of a

The word "raised" is usually not used neither is the word "power" used so 3^{5} is typically pronounced "three to the fifth" or "three to the five.

**4.2.2 Negative integer exponents**: A negative exponent means to **divide** by that number of factors **instead of multiplying**. So 3^{-3} is the same as 1/(3^{3}), therefore in general term we can write it as

a^{-n} = (1/a^{n})

example: 5^{-3} can be written as

5^{-3}= (1/ 5^{3}) = 1/(5x5x5) = 1/125

Note* n≠ 0, n = 0 is not defined for a^{-n}

**4.2.3 Exponents of zero and one:** We know that anything**to the power 0 **is 1.

**Proof**:

Assume a^{0}. By the division rule we know that,

a^{n}/a^{n} = a^{(n-n)} = a^{0}. (i)

But anything divided by itself is 1, so

a^{n}/a^{n}= 1.

If a^{n}/a^{n} is equal to 1 and from (i) we have a^{n}/a^{n} equals to a^{0}, then 1 must equal a^{0}. Symbolically,

a^{n}/a^{n} = a^{(n-n)} = a^{0} = 1

we created a fraction to figure out a0. But division by 0 is not allowed, so our evaluation is defined for anything to the 0 power * except* zero itself.

**4.3.1 Powers of ten**: In the decimal number system, integer powers of 10 are written as the digit 1 followed by a number of zeroes, depending on the sign and magnitude of the exponent.

For example, 10^{3} = 1000

10^{-4} = 0.0001.

Exponentiation with base 10 is used in scientific notation to describe large or small numbers.

**For example**: 3452178 can be written as

34.52178 x 10^{5}

SI prefixes based on powers of 10 are also used to describe various small and large quantities.

**For example**,

the prefix kilo means 10^{3} = 1000, so

1 km= 1000m.

1 kg = 1000gm

**4.3.2 ****Powers of two****: **The positive powers of 2 have a great importance in computer science because there are 2^{n} possible values for an n-bit variable.

Powers of 2 are important in set theory since a set with n members has a power set, or set of all subsets of the original set, with 2^{n} members.

The negative powers of 2 are commonly used, and the first two have special names: half, and quarter.

In the base 2 (binary) number system, integer powers of 2 are written as 1 followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, two to the power of three is written 1000 in binary.

**4.3.3 ****Powers of one**: The integer powers of one are one: 1^{n} = 1.

**4.3.4 Powers of zero****: **If the exponent is positive, the power of zero is zero: 0^{n} = 0, where n > 0.

If the exponent is negative, the power of zero (0^{n}, where n < 0) is undefined, because division by zero is implied.

**4.3.5 ****Powers of minus one****: **If n is an even integer, then (-1)^{n} = 1. If n is an odd integer, then (-1)^{n} = −1. Because of this, powers of −1 are useful for expressing alternating sequences.

**4.4****Laws of exponent**:

i) a^{m+n} = a^{m} . a^{n}

This identity has the consequence

ii) a^{m-n} = a^{m} = 1

a = a^{n-m}

for *a* ≠ 0,

iii) (a^{m})^{n} = a^{m.n}

Another basic identity is

iv) (a.b)^{m} = a^{m} . b^{m}

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