Algebra Basics: Exponents In Algebra - Math Antics - Free Educational videos for Students in K-12

Algebra Basics: Exponents In Algebra - Math Antics - By Mathantics

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00:03	Uh huh . Hi , I'm rob . Welcome to
00:07	Math Antics . We've already learned a little about how
00:10	exponents and roots are used in arithmetic and now it's
00:13	time to learn the basics of how they're used in
00:15	algebra . As you know , one of the main
00:18	differences between arithmetic and algebra is that algebra involves unknown
00:23	values and variables in arithmetic . You might have the
00:26	exponent for squared , but in algebra you're more likely
00:30	to see the exponent X squared and when it comes
00:33	to roots instead of seeing the square root of 16
00:36	, you might see the square root of X .
00:39	Of course . One of the main goals in algebra
00:41	is to figure out what those unknown values are and
00:44	we're going to learn a bit about how to do
00:46	that in a minute . But first we're going to
00:48	learn something about exponents by looking at an important pattern
00:51	in algebra . It's the pattern formed by the expression
00:55	X to the 10th power . Where in is any
00:58	integer in this expression X could be any number but
01:02	in can only be an integer . And to keep
01:05	things simple in this video we're only going to consider
01:07	non negative integers . That is will limit in to
01:11	be this set of numbers 0123 and so on .
01:15	If n is zero then we have X to the
01:18	zero with power . If n is one then we
01:21	have X to the first power . If n is
01:24	two , then we have X to the second power
01:26	or X squared if n is three then we have
01:29	X to the third power or X cubed . And
01:32	we could keep on going with this pattern . X
01:34	to the fourth . X to the fifth to infinity
01:37	. Okay . But what do these exponents mean ?
01:40	Well X squared is pretty easy to understand . We
01:43	know from our definition of exponents that X squared would
01:46	be the same as X times X . We also
01:49	know that X cubed would be X times X times
01:52	X and going up to higher values of in would
01:55	just means multiplying more exes together . But what about
01:58	X to the first power ? Well if X to
02:01	the second power means multiplying two exes together , then
02:05	X to the first power should mean multiplying one X
02:08	together . Which sounds kind of funny when we say
02:10	it like that . But as you can see that
02:13	pattern makes sense . X to the first power would
02:16	just be X . And that helps us see an
02:18	important rule about exponents . Any number raised to the
02:22	first power is just itself . This rule or property
02:26	is similar to the identity property of multiplication that says
02:29	any number multiplied by one is just itself . Okay
02:33	, so X to the first power makes sense .
02:35	But what about X To the zero with power ?
02:38	Does that mean no excess multiplied together ? That seems
02:41	even stranger . And the rule about the zeroth power
02:44	may surprise you . It seems like X to the
02:47	zero power should be zero . But it's actually one
02:50	which will make a lot more sense if we modify
02:52	our pattern a little . Do you remember that because
02:55	of the identity property of multiplication , there is always
02:58	a factor of one in any multiplication problem Or is
03:02	the same as one times 4 ? five is the
03:05	same as one times 5 and so on . Well
03:09	, that means we can also include a factor of
03:11	one . And our pattern of exponents X to the
03:13	first is one times X . X to the second
03:17	is one times X times X . X to the
03:19	third is one times X times X times X and
03:22	so on . And if we continue that pattern ,
03:25	the other direction , you'll see that there will be
03:27	a one left there , even when all the exes
03:29	are gone . So now , you know , another
03:31	important rule about exponents , Any number raised to the
03:35	zeroth power is just one . Knowing these rules about
03:39	exponents is important in algebra and will help us when
03:42	we talk about polynomial is in the next video .
03:44	Before the rest of this video , we're going to
03:46	learn how to solve some really basic algebraic equations that
03:49	involve exponents and roots . Let's start off with this
03:53	equation , The square root of X equals three .
03:56	How do we solve for X . And this equation
03:58	, in other words , how do we figure out
03:59	the value of X without just guessing the answer ?
04:02	Well , we know that the key to solving an
04:05	algebraic equation is to get the unknown value all by
04:08	itself on one side of the equal sign . And
04:11	you might be thinking that in this equation , the
04:13	X looks like it's already by itself after all ,
04:16	there's no other numbers with it . But getting an
04:19	unknown by itself means we need to isolate it from
04:21	any other numbers and operators so that it's completely by
04:25	itself in this equation , that means we need to
04:28	somehow get rid of the square root sign that the
04:30	X is under . Ha need to get rid of
04:33	that pesky square root sign . Do you ? Let's
04:36	see . I'll just use my magic wand and that
04:41	usually works . I know . Uh huh . This
04:51	is gonna be harder than I thought , wow .
04:55	1 2 . Whoa , hold on . Now that
04:59	seems a bit extreme and it won't even help .
05:02	I mean this is a math operation . And to
05:04	get rid of a math operation , you need to
05:06	use its inverse operation . Well , I was gonna
05:09	try that next in the video called exponents and square
05:14	roots . We learn that . Exponents and roots are
05:17	inverse operations . If we want to undo an exponent
05:20	, we need to use a route and if we
05:22	want to undo a route we need to use an
05:23	exponent . So in this equation to undo the second
05:27	route or square root of X , we're going to
05:30	need to raise it to the second power or square
05:33	it . If we square the square root of X
05:36	, those operations will cancel out and we'll be left
05:40	with just X . But why does that work ?
05:43	Well , you can see why it works . If
05:45	you remember what the square root of X really means
05:48	, The square root of X is a number that
05:50	we can multiply together twice to get X . For
05:54	example , the square root of four is too ,
05:57	because if you multiply two times two , you get
06:00	four . So since the square root of four is
06:03	the same as to , We could also just say
06:06	that the square root of four times the square root
06:08	of four is 4 . And do you see how
06:11	the square root of four times the square root of
06:14	four is the same as the square root of four
06:17	squared ? And this is true for any number ,
06:20	which is why squaring the square root of X just
06:23	leaves us with X . The exponent and the route
06:26	operations cancel each other out . Okay , so we
06:30	can undo the square root by squaring that side of
06:33	the equation . But remember to keep our equation and
06:36	balance , we need to do the same thing to
06:39	both sides . So we need to square the three
06:41	also three squared is three times three , which is
06:45	nine there . By squaring both sides of the equation
06:49	. We changed it into X equals nine . We
06:51	solve for X . That was pretty easy . Let's
06:54	try solving another simple problem with the root . This
06:57	one is the cube root of x equals five .
07:01	Just like before . We need to get X all
07:03	by itself by undoing the route . But since it's
07:07	a cube root this time , we can't undo that
07:09	by squaring both sides . Instead , we need to
07:12	cube both sides . You always need to undo a
07:15	route with the corresponding exponent , third root , third
07:19	power , fourth root , fourth power and so on
07:23	. So to solve this equation , we need to
07:25	raise each side of the equation to the third power
07:28	. On the first side , the operations cancel leaving
07:31	X all by itself . On the other side we
07:34	have five to the third power , which is five
07:37	times five times five or 125 . So x equals
07:42	125 . All right . So that's how you saw
07:46	very simple one step equations with roots . What about
07:49	simple equations that have exponents instead of roots like this
07:53	? One X squared equals 36 . Again , we
07:57	need to get the X all by itself , which
07:59	means we need to deal with the exponent on this
08:01	side of the equation . How do we undo an
08:03	exponent , yep . We use a route since the
08:07	X is being squared . If we take the square
08:09	root of X squared , the operations will cancel out
08:13	leaving X all by itself . But why does that
08:16	work ? Well , think for a minute about what
08:18	the square root of X squared would mean ? It
08:22	means that you need to figure out what number you
08:24	can multiply together twice in order to get X squared
08:28	. But that's easy . X times X is X
08:31	squared . So that means that the square root of
08:34	X squared is just X . So to solve this
08:37	equation , we take the square root of both sides
08:40	of the equation to keep things in balance on the
08:42	first side , the operations cancel out leaving X all
08:46	by itself . And on the other side We have
08:49	the square root of 36 , which is six .
08:52	So the answer to this problem is x equals six
08:56	. Well , that's half of the answer anyway .
08:59	This problem is actually a little more complicated than it
09:01	looks at first . Thanks to negative numbers . Do
09:05	you remember in our video about multiplying and dividing integers
09:09	? We learned that if you multiply two negative numbers
09:11	together , the answer is actually positive . That turns
09:15	out to be really important when it comes to roots
09:19	because it means that there's often more than one answer
09:22	. For example , we know that the square root
09:24	of 36 is six because multiplying six times six gives
09:28	us 36 . But because of that rule about negative
09:32	numbers negative six times negative six is also 36 .
09:37	So it would be just as correct to say that
09:39	the square root of 36 is -6 . So which
09:43	is it is the square root of 36 6 or
09:46	-6 ? The answer is both . This is an
09:50	example of a simple algebraic equation that has two solutions
09:55	. X could be six or X could be negative
09:57	six . X can't be both six and negative six
10:00	at the same time . But you could substitute either
10:03	value into the equation and it would make the equation
10:06	true . So in algebra , when we have a
10:09	situation like this where the answer could be positive or
10:12	negative , use a special plus or minus sign that
10:15	looks like this X equals plus or -6 . And
10:19	we use it when we're finding even roots of a
10:22	number . Since we know that the answer could be
10:24	positive or negative . But what about odd routes like
10:28	the cube root of a number ? Like what if
10:31	we have to solve the equation X cubed equals 27
10:35	to solve this equation for X . We need to
10:37	take the cube root of both sides . On the
10:40	first side of the equation , the cube root will
10:42	cancel out the cube operation that's being done to X
10:45	. Leaving X all by itself . And on the
10:48	other side we need to figure out the cube root
10:50	of 27 . Using a calculator . Or just by
10:53	knowing about the factors of 27 . We see that
10:56	the cube root of 27 is three Because three times
11:00	3 times three is 27 . So in this equation
11:03	we know that x equals three . But what about
11:06	negative numbers ? Is X equals negative three . Also
11:09	a valid solution to this equation , nope . And
11:12	here's why if you multiply negative three times negative three
11:16	times negative three , the answer would be negative .
11:18	27 not 27 . So the cube root of 27
11:22	is three but not negative three . In this case
11:26	there is only one solution . Alright . In this
11:29	video we learned two important rules about exponents . We
11:33	learned that any number raised to the zeroth power equals
11:36	one and that any number raised to the first power
11:39	is just itself . We also learned how to solve
11:43	very simple one step equations involving exponents and roots .
11:47	Since their inverse operations to undo a route , you
11:50	use its corresponding exponent and to undo an exponent you
11:54	use its corresponding route . Of course there's a lot
11:57	more to learn about exponents in algebra . But those
12:00	are the basics and to make sure you really understand
12:02	them , it's important to practice by doing some exercise
12:05	problems , as always . Thanks for watching Math Antics
12:08	and I'll see you next time learn more at Math
12:12	Antics dot com .

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