MATHEMATICS
TOPICS-
EXPONENTS AND POWERS
Exponents and Powers
An exponent or power is a
mathematical representation that indicates the number of times that a number is
multiplied by itself.
If a number is
multiplied by itself m times, then it can be written as: a × a × a × a × a...m
times = am
Here, a is called the base, and
m is called the exponent, power or index.
Important Points
· Numbers raised to the power of two are called square numbers.
· Numbers raised to the power of three are called cube numbers.
· Negative numbers can also be written using exponents.
· If an = b, where a and b are integers and n is a natural number, then an is called the exponential form of b.
· The value of an exponential number with a negative base raised to the power of an even number is positive.
· If the base of two exponential numbers is the same, then the number with the greater exponent is greater than the number with the smaller exponent.
Exponantial form of a number:
Since 625 = 5 × 5 × 5 × 5 = 54. We say 54 is the exponential form of 625. It is also ead as 5 raised to
power 4. Standard form
A number can be expressed as a decimal number between 1.0 and 10.0, including 1.0, multiplied by a power of 10. Such a form of a number is known as its standard form.
For Example: 625 = 6.25 × 102
Laws of
Exponents
1. Product Law: When numbers with the same base are multiplied, the power of the product is equal to the sum of the powers of the numbers keeping the base same. More precisely if m and n are whole numbers then,
𝐚𝐦 × 𝐚𝐧 = 𝐚𝐦 + 𝐧
2. Quotient
Law: When numbers with the same base are divided, then the power
of the quotient is equal to the difference
between the powers of the dividend and the divisor keeping the base same. That
is, if a is a non-zero integer, and m and n are whole numbers then,
𝐚𝐦 ÷ 𝐚𝐧 = 𝐚𝐦 – 𝐧
3. Power Law: when a number in the index is raised to another index, the base is raised to the product of the two indices. That is,
(𝐚𝐦)𝐧 = 𝐚𝐦×𝐧
More about Indices
a) (𝒂𝒃) = 𝒂𝒎 × 𝒃𝒎
)
b) (𝒂
𝒎
𝒃
𝒂𝒎
=
𝒃𝒎
c) 𝒂𝟎 = 𝟏
d) If 𝒂 ≠ 𝟎 then, 𝒂−𝒎= and 𝟏
= 𝒂𝒎
𝒂𝒎 𝒂−𝒎
Exponents and Powers
An exponent or power is a mathematical representation that indicates the number of times that a number is multiplied by itself.
An exponent or power is a mathematical representation that indicates the number of times that a number is multiplied by itself.
If a number is
multiplied by itself m times, then it can be written as: a × a × a × a × a...m
times = am Here, a is
called the base, and m is called the
exponent, power or index.
Numbers raised to the power of two are called square numbers.
Square numbers are also read as two-square, three-square, four-square, five-square, and so on. Numbers raised to the power of three are called cube numbers.
Cube numbers are also read as two-cube,
three-cube, four-cube, five-cube, and so on. Negative
numbers can
also be written
using exponents.
If an = b, where a and b are integers and n is a natural number, then an is called the exponential form of b.
The factors of a product can be expressed as the powers of the prime factors of 100. This form of expressing numbers using exponents is called the prime factor product form of exponents. Even if we interchange the order of the factors, the value remains the same.
So a raised to the power
of x multiplied by b raised to the power of y, is the same as b raised to the
power of y multiplied by a raised to the power of x.
The value of an exponential number with a negative base raised to the power of an even number is positive.
If the base of two exponential numbers is the same, then the number with the greater exponent is greater than the number with the smaller exponent.
A number can be expressed as a decimal number between 1.0 and 10.0, including 1.0, multiplied by a power of 10. Such a form of a number is known as its standard form.
Laws of Exponents
When numbers with the same base are multiplied, the power of the product...
When numbers
with the same base are multiplied, the power of the product
is equal to the sum of the powers of the numbers.
More precisely if m and n are whole numbers
then, am × an = am + n
When numbers
with the same base are divided, then the power of the quotient is equal to the difference between the powers
of the dividend and the divisor. That is, if a is a non-zero
integer, and m and n are whole numbers
then, am ÷ an = am – n
The other laws of exponents are as follows:
1) where > are non - zero integers, and is a whole number.
2) where are non - zero integers, and is a whole number.
3) am ÷ an = am – n,where > a is a non-zero integer,
and m and n are whole numbers.
4) The value
a0 = 1 ,where > a is a non-zero
integer.
5) where > a is a non - zero integer, and m and n are whole numbers.