hi welcome to math antics in our last
video called intro to exponents we
learned that exponents also called
indices are a special type of math
operation in this video we're going to
expand on what we know about exponents
by learning about their inverse
operations which are called roots that's
kind of a strange name for a math
operation but it'll make more sense in
just a minute first let's review what we
mean by inverse operations in the video
called what is arithmetic we learned
that inverse operations are pairs of
math operations that undo each other for
example you can undo addition by doing
subtraction so addition and subtraction
are inverse operations likewise you can
undo multiplication by doing division so
multiplication and division are inverse
operations as I mentioned exponents have
inverse operations also there are
operations that can undo them and those
operations are called roots to see how
roots and exponents work together to
undo each other let's look at the simple
exponent 4 to the second power or 4
squared previously we learned that this
is the same as 4 times 4 which equals 16
doing this exponent meant going from 4
squared - 16 so now if we want to undo
that with the root operation that
involves starting out with 16 and then
somehow getting back to the 4 which is
being raised to the second power and do
you remember what that part of the
original exponent is called yep it's
called the base so doing a root
operation is going to give us the base
as our answer and that helps us
understand its name a little better the
words base and root have a similar
meaning especially if you think of a
tree the root is at the base of a tree
and that can help you remember how root
operations work with the root operation
you start with the answer of an exponent
and try to figure out what the base of
that original exponent is ok but how do
we actually do that how do we use a root
operation to go back
and figure out the base of the original
exponent well for starters we need to
know about a special math symbol that
looks like this and you guessed it it's
called the root sign whoa dude that math
symbol looks totally radical dude it's
it's like that division thingy only way
cooler ah yes
that reminds me the root sign is often
referred to as the radical sign and
mathematicians use that term even before
surfers did and yes it does look similar
to the division sign so it's really
important not to get them confused the
root or radical sign is different from
the division sign because instead of
having a curved front its front shape is
like a check mark the number that you
want to take the root of goes under the
sign like this so when you see a number
under a root or radical sign like this
you know you need to figure out the base
of the original exponent in this case
you need to figure out what number you
can multiply together a certain number
of times to get 16 ah but there's the
catch how many times the answer we get
from taking the root will depend on how
many times that number would be
multiplied together but that would
depend on the original exponent so how
do we know what that number is simple
the root symbol tells us the root symbol
actually includes the original exponent
in it what you don't see it oh that's
because I didn't draw it in yet and
later in this video you'll understand
why so let's put a little two right here
above the check mark part of the root
symbol and that too tells us that we
need to figure out what number or base
could be multiplied together two times
in order to get 16 and if you remember
your multiplication table or if you just
look at our original example here you'll
know that the answer to that is 4 now do
you see how the root operation is the
inverse of the exponent operation when
doing the exponent we asked what do we
get if we multiply four together two
times and the answer was 16 but when we
did the root operation we asked what
number could we multiply together two
times to get 16 and the answer was four
great now that you understand how
exponents and roots are related we're
going to look closer at how rude
operations work to do that we're gonna
change our root problem slightly let's
change the little two into a little for
the first root was asking us to figure
out what number we could multiply
together two times to get 16 but this
new root is asking us to figure out what
number we could multiply together four
times to get 16 that's a bit trickier
huh can you think of a number like that
yep
the answer is two because if you
multiply it four twos together two times
two times two times two you get 16 so
the second root of 16 is 4
but the fourth root of 16 is 2 both
those roots were pretty easy to figure
out right but unfortunately figuring out
roots and math can be much harder for
example what if we add this problem
instead root 3 of 16 that means we need
to figure out what number we can
multiply together three times to get 16
can you think of a number like that no I
can't either and unfortunately it's not
easy to calculate what that number would
be remember even though this looks a
little bit like the division symbol this
is not just division you can't just
divide 16 by 3 to get the answer roots
are not the same as long division so how
do we calculate a root like this well
there are special algorithms that you
can use to calculate just about any root
but they're kind of complicated so we'll
save those for a future video instead
I'm going to use a special root function
on my calculator to get the answer and
on my calculator the button for that
root function looks like this to use it
I first enter the number that I want to
take the root of which is 16 next I hit
the root function button and then I
enter 3 so it knows that I want the
third root of 16 last I hit the equal
sign and voila the answer is two point
five one nine eight four two and the
decimal digits just keep on going
forever Wow
see what I mean about roots being hard
to figure out this is a really
complicated decimal number
and you may even wonder if it's the
right answer well let's check based on
what we know about exponents and roots
if we multiply this decimal number
together three times we should get
sixteen right but to make it easier to
check let's just round the number off to
two decimal places let's make it two
point five two if we multiply two point
five two together three times in other
words if we take two point five two to
the third power
we'll get sixteen point zero zero three
well that's almost right it's really
close to 16 isn't it the reason it's not
exactly 16 is that we rounded the number
off which made it less accurate but the
more decimal digits we use the closer
we'll get to 16 in math the vast
majority of roots are complicated
numbers like this and they're hard to
figure out unless you use a calculator
or a special algorithm that's the bad
news but the good news is that most of
the time the roots you'll be asked to do
in your homework or on tests are the
easy ones the ones that have nice whole
number answers and usually you'll only
be asked to find the second or third
roots of numbers do you remember in the
last video we learned that 2 & 3 are the
most common exponents so common in fact
that they even had special names raising
a number to the second power was called
squaring it and raising a number to the
third power was called cubing it well
it's the same with roots since the roots
2 & 3 are the most common they get
special names also the second root is
called the square root and the third
root is called the cube root
in fact the square root is so common
that it's basically the default route
and its symbol even gets special
treatment do you remember that when I
first showed you the root symbol I left
out the index number that tells you what
root to find well whenever that number
is left out you can just assume that
it's 2 in other words the root symbol
with no index number is always the
square root so if you want someone to
find a different route like cubed or 4th
or 5th then you need to include that
number so they know which route to find
and even those square roots are the most
they're not always easy to find most are
still going to be big long decimal
numbers except for the perfect squares
it's easier to find the square roots of
the perfect squares because their
answers can be found using the
multiplication table on the
multiplication table have you ever
noticed that all the answers to problems
where the same numbers being multiplied
together or on the diagonal of the table
in other words 2 times 2 equals 4 3
times 3 equals 9 4 times 4 equals 16 5
times 5 equals 25 6 times 6 equals 36
and so on well those numbers are called
the perfect squares because they're the
answers you get when you square a whole
number and that means if you take the
square root of a number along that
diagonal you get a nice whole number as
your answer the square root of 4 is 2
the square root of 9 is 3 the square
root of 16 is 4 the square root of 25 is
5 and so on see what I mean
those roots are really common and
they're also easy to figure out if you
know your multiplication facts so if
you're new to exponents and roots
learning the perfect squares is the
place to start once you understand how
those exponents and roots work you'll be
ready to figure out tougher problems all
right so now you know how exponents are
related to roots there are inverse
operations and they undo one another and
you also know that just like 2 & 3 are
the most common exponents the square
root and the cube root are the most
common roots we'll also know that
finding roots is usually not very easy
that's important to know so you don't
get discouraged if you feel like it's
hard to figure out what a certain root
is you're not alone we think it's hard
to and would normally just use a
calculator to find them the good news is
that some roots are easy to find like
the perfect squares so be sure to focus
on learning them first and remember to
get good at math you need to actually
practice what you learned from watching
videos so be sure to do some exercise
problems and as always thanks for
watching math antics and I'll see you
next time
learn more at math antics calm