You are not currently logged in. Please create an account or log in to view the full course.
Indices – B1
- About
- Transcript
- Cite
Algebra and Functions I – AQA (7356)
In this course, Dr Niki Kalaydzhieva (UCL) explores algebra and functions, covering Topics B1, B2, B3, and B6 in the AQA AS Mathematics (7356) Specification. In the first mini-lecture, we recall how to used indices and the index laws (Topic B1). In the second mini-lecture, we work with fractional and negative indices (Topic B1). In the third mini-lecture, we practice expanding brackets and factorising (Topics B3, B6). In the fourth mini-lecture, learn how to factorise quadratic expressions (Topics B3, B6). In the fifth mini-lecture, we factorise cubic and quartic expressions before learning the difference of squares method (Topic B3). In the sixth mini-lecture, we introduce surds (Topic B2). In the seventh mini-lecture, we learn how to rationalise the denominator (Topic B2).
Indices – B1
In this mini-lecture, we introduce Topic B1 by recalling index laws. In particular, we consider: (how to use index notation to simplify expressions; (ii) the laws of indices and examples using these laws; and (iii) why the index laws work.
Hi, everyone. My name is doctor Nikikalejieva.
00:00:05And I'm a lecturer at University College London in
00:00:08the Department of Mathematics.
00:00:12Welcome to this algebra and function course.
00:00:14The first lecture is about algebra revision.
00:00:19We're going to start with indices then we're going to
00:00:23move on to expanding and factorizing expressions and
00:00:26finally we're going to talk about thirds. So let's get into
00:00:30indices.
00:00:35We know that if adding the same number or letter to itself
00:00:37multiple times,
00:00:41then we can rewrite this expression using multiplication.
00:00:42So for example, two plus two plus two can be rewritten as
00:00:46three times two. Similarly, if we have x plus x, we
00:00:51can write this as two times x. So we can do a very
00:00:55similar thing when we have the same expression being
00:01:00multiplied to itself times,
00:01:04but we use an index instead or indices if
00:01:06we're meaning in plural.
00:01:12So for example, if we have x times x, we can write this
00:01:14as x squared. This little thing on top is the index
00:01:19or the power or the exponent
00:01:23and it tells us how many times do we multiply the basis
00:01:26element to itself. And the x is called the base. It tells
00:01:30us what letter or number or even an expression is being
00:01:35multiplied by itself many times.
00:01:40So here is a quick example.
00:01:44Suppose we have two times two times two, We can rewrite this
00:01:47as two to the power of three,
00:01:51and we have a special word for that expression.
00:01:53We say it's two cubed.
00:01:56If we look at three y times three y then that
00:01:59can be rewritten as three y to the power of two. And we
00:02:03also have a special word for when we're putting things to
00:02:08the power of two. We use the word square
00:02:11You might also be familiar with the laws of indices. So we
00:02:15use these to make complex problems involving powers much simpler.
00:02:20They're the rules that tells us how to multiply and divide indices.
00:02:26Just a quick reminder of what these laws are. We have a to
00:02:32the m times a to the n is eight the m plus n, a
00:02:36to the m divided by a to the n is a to the m minus n,
00:02:41and a to the m to the power of n is a to the
00:02:47power of m times n. And the last one is a
00:02:51times b all to the power of n is a to the n
00:02:56times B to the n.
00:03:00We can see these in actions in some examples.
00:03:03Suppose we have two times r squared times three
00:03:07times r to the power of five.
00:03:11The first thing we can do is we can put all the numbers
00:03:14together and all the r's together.
00:03:17So we can rearrange this expression and rewrite it as
00:03:18two times three times r squared times r to the power of five.
00:03:21Two times three is equal to six, then r squared times r to
00:03:26the five, we can use the index laws to re write as r to the
00:03:31two plus five, so overall we end up with six times
00:03:34r to the seven.
00:03:39Okay? Let's now look at the second example,
00:03:41six x to the five divided by three x squared.
00:03:45We can rewrite this
00:03:50using the fractional notation as six x to the
00:03:53five divided by three x squared.
00:03:57We can take the numbers separately and the x's from exes.
00:04:00So we have six over three times x to the five divided
00:04:05by x squared, which is
00:04:09to times x to the five minus two or two
00:04:13times x cubed.
00:04:17In the final example we're going to look at,
00:04:21is three x squared, all cubed
00:04:23divided by x to the fourth. So this is a bit more complex we
00:04:27have a few of the index laws kind of altogether.
00:04:31So the first thing we're going to do,
00:04:35we're going to use law number four and expand the brackets.
00:04:37So that gives us three cubed times X squared to the power of three
00:04:41and all divided by x to the four.
00:04:48We once again put the numbers to one side and all of the
00:04:51expressions involving x to the side and we have three cubed times
00:04:54x to the power of two times three using law number three
00:05:00divided by to the four.
00:05:03Three cubed is twenty seven and x to the power of two times
00:05:06three is x to the six, dividing this by x to the four, gives us
00:05:10twenty seven times x to the six minus four, which ends
00:05:14up being twenty seven times x
00:05:17Okay. But why do these laws work the way they do? Well,
00:05:23gonna give you a couple of examples to help you understand.
00:05:28Suppose we have a to the five times a squared.
00:05:33We can rewrite eight to the five as a times a times a times
00:05:37a times a, so this is five lots of A. And we multiply this, but
00:05:41a squared, which we can rewrite as a times a.
00:05:45If we now look at the number of a's we have,
00:05:49we have exactly five plus two which is seven eighth.
00:05:51Therefore,
00:05:56a to the five times eight to two is eight to the seven which
00:05:57is eight to the five two. And this works in general.
00:06:00If we have a to the m times a to the n, we can rewrite a to
00:06:04the m back into the multiplication forms so we have
00:06:08a multiplied by itself m times and then a to the n has a
00:06:12multiplied by itself n times.
00:06:16So overall we have a multiplied by itself n plus n times.
00:06:18So the final result is a to the power of n plus n. I mean,
00:06:22the same works for division.
00:06:26Suppose we have a to the five
00:06:28divided by a squared,
00:06:32we can write this in the fractional form and expand using
00:06:33application.
00:06:37So on in the numerator we're gonna have a times a times a
00:06:38times a times a so that's five lots of a and in the
00:06:41denominator we have a times We can cancel out the two a's in
00:06:44the denominator with two of the a's in the numerator,
00:06:49which gives us three lots of a, which is a cubed. So once
00:06:51again, a to the five divided by a squared is a to the power
00:06:56of five minus two which is a cubed.
00:07:00And then you can see this in general as well.
00:07:04And the final example I'll show you is how a squared
00:07:09or the power of a cube works.
00:07:13Well if we expand the power from the outside in,
00:07:15we have a squared squared multiplied by its
00:07:18of three times.
00:07:22So that should be a squared times a squared times a squared
00:07:23and we can now expand each one of those x a squareds as a times a.
00:07:27You can see this written in the brackets.
00:07:32So what happens now is that we have three times
00:07:36two lots of a. So that's eight to the power of two times
00:07:40three, which is eight to the six.
00:07:44
Cite this Lecture
APA style
Kalaydzhieva, N. (2023, July 18). Algebra and Functions I – AQA (7356) - Indices – B1 [Video]. MASSOLIT. https://massolit.io/courses/algebra-and-functions-i-aqa-7356
MLA style
Kalaydzhieva, N. "Algebra and Functions I – AQA (7356) – Indices – B1." MASSOLIT, uploaded by MASSOLIT, 18 Jul 2023, https://massolit.io/courses/algebra-and-functions-i-aqa-7356
Lecturer