Promo: What is 0 raised to the power of 0? Video Lecture | Sets and Functions - JEE

what is zero raise to the power of zero
let's see what is zero raise to other
powers zero raised to the third power is
zero zero raised to the second power is
zero zero raised to the first power is
also zero
following the positive exponents it
seems like the zero raised to the power
of zero should also equal zero but to
zero to the negative second power is
undefined the same is true for zero to
the negative first power following the
negative exponents it seems like the
zero to the power of 0 should be
undefined
other numbers raised to zero equal one
this pattern implies that 0 raised to 0
should be 1 it looks that there is no
obvious value of 0 raised to 0 which is
true depending on the context we are
working in one answer may be better than
the others
in algebra combinatorics or set theory
the generally agreed upon value is 0
raised to 0 equals 1 let's have a look
at the binomial theorem 4x equals 0
this simplifies to 1 equals 0 raised to
the power of 0 times 1 therefore for the
binomial theorem to be valid for x
equals 0 0 raised to the power of 0
should be 1 we can view a race to the
power of B as the number of tuples
containing the elements that can be
chosen from a set of a elements for
example we can view 3 raised to the
power of 2 as the number of tuples
containing 2 elements that can be chosen
from a set of 3 elements
similarly 3 raised to the power of 0 is
the number of tuples containing 0
elements that can be chosen from a set
of 3 elements which equals 1 0 raised to
the power of 0 is the number of tuples
containing 0 elements that can be chosen
from a set of 0 elements which is also 1
1 is the only definition consistent with
this interpretation of exponentiation
in differential calculus the power rule
is not valid for N equals one at x
equals zero unless we define zero raised
to the power of zero to be one however
as discussed earlier depending on the
context one may not always be the best
answer for example let's look at some
limits limit of a function at a point a
is the value that a function approaches
as the input approaches a we are
interested in the limits of the form 0
raised to 0 when x equals 0 the limit of
X raised to X as X approaches 0 from the
right is 1 this implies if we want X
raise to X to be right continuous at 0
we should define 0 raised to the power
of 0 to be 1 the limit of X raised to 0
as X approaches 0 is 1 this again
suggests that 0 raised to 0 is 1 but
there are other limits of the same form
with different values
the limit of 0 raised to X as X
approaches 0 from the right is 0 the
limit of 0 raised to X as X approaches 0
from the left is undefined
the limit of X raised to 1 over natural
log X as X approaches 0 from the right
is e so the value of the limits of the
form 0 raised to the power of 0 may
differ with the functions we are dealing
with therefore when dealing with limits
0 raised to 0 is an indeterminate form
or undefined
so what is zero raised to the power of
zero depending on the context where it
occurs we might wish to substitute it
with one indeterminate or undefined
often one is the useful choice this
convention allows us to extend
definitions in different areas of
mathematics that would otherwise require
treating zero is a special case however
when dealing with limits indeterminate
form or undefined as the more obvious
choice