Math Assignment Help With Exponents And Operation On Exponents

Chapter 4. Exponents And Operation On Exponents

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.

aⁿ = a x a x a x……. a

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

example: a=4; n=2

aⁿ = 4² = 4 x 4 = 16

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

4.2 Integer exponent : the exponentiation of integers is based on basic algebra.

4.2.1 Positive integer exponents:

n= positive integer

a² = a x a is called the square of a

a³ = 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 3⁵ 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/(3³), therefore in general term we can write it as

a^-n = (1/aⁿ)

example: 5^-3 can be written as

5^-3= (1/ 5³) = 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 anythingto the power 0 is 1.

Proof:

Assume a⁰. By the division rule we know that,

aⁿ/aⁿ = a^(n-n) = a⁰. (i)

But anything divided by itself is 1, so

aⁿ/aⁿ= 1.

If aⁿ/aⁿ is equal to 1 and from (i) we have aⁿ/aⁿ equals to a⁰, then 1 must equal a⁰. Symbolically,

aⁿ/aⁿ = a^(n-n) = a⁰ = 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 Powers:

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, 10³ = 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 10⁵

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

For example,

the prefix kilo means 10³ = 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 2ⁿ 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 2ⁿ 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: 1ⁿ = 1.

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

If the exponent is negative, the power of zero (0ⁿ, 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.

1. 4.4 Laws of exponent:

i) a^m+n = a^m. aⁿ

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

Email Based Assignment Help in Exponents And Operation On Exponents

To submit Exponents And Operation On Exponents assignment click here.