Lesson video

Hi everyone! My name is Ms Coo.

And I'm really happy that you're learning with me today.

We are going to be looking at arithmetic procedures: index laws, a really interesting lesson.

I hope you enjoy it.

I know I will.

So let's make a start.

Hi everyone and welcome to this lesson on laws of indices, negative and zero exponents under the unit, Arithmetic procedures: index laws.

And by the end of the lesson, you'll be able to use laws of indices with negative and zero exponents.

We're gonna look at some key words.

First of all, just to recap that numbers that have been multiplied by themselves as a repeated number of times can be expressed using a base and an exponent.

For example, two, multiply by two, multiply by two is written as two to the power of three.

We know the three is the integer exponent and identifies how many times the base has been multiplied by itself.

And the two, well the two is the base.

Well, this represents the base and is the number or sometimes term that's been multiplied by itself.

An alternative word for exponent is index and you'll be hearing this word index or indices all up through the lesson.

Now we'll also be using the word reciprocal.

And a reciprocal is the multiplicative inverse of any non-zero number.

And any non-zero number multiplied by its reciprocal is equal to one.

For example, five and one-fifth, these are reciprocals of one another because five multiply by one-fifth is equal to one.

Two-thirds and three over two, these are reciprocals of one another because two-thirds multiply by three over two is equal to one.

Four and 0.

4 are not reciprocals of one another because four multiply by 0.

4 does not give us one.

Today's lesson will be broken into three parts.

First of all, we'll be exploring negative exponents.

Then we'll be looking at the exponents of zero and then we'll be looking at fractional basis.

So let's make a start, exploring negative exponents.

Now we have seen negative exponents sometimes called negative indices in a place value chart.

For example, here you can see using our place value chart are powers of 10 given in index form.

What do these negative exponents mean on the place value chart? Well, let's have a look at the index form and the fractional form so we can see these equivalents where we know 10 to the zero is one over one.

We know 10 to the negative one is one-10th.

10 to the negative two is one over 100.

10 to the negative three is one over 1000.

And 10 to the negative four is one over 10,000.

So converting into fractional form and index form, we know all of these are equivalent.

So that means we can write using index and fractional form.

10 to the zero is one over 10 to the zero.

10 to the negative one is one over 10.

10 to the negative two is one over 10 squared.

10 to the negative three is one over 10 cubed.

And 10 to the negative four is one over 10 to the power of four.

So therefore, we have these equivalents.

10 to the negative one is one over 10.

10 to the negative two is one over 10 squared.

10 to the power of negative three is one over 10 cubed and 10 to the power of negative four is one over 10 to the four.

I want you to have a little think.

What do you think the equivalent would be for this one? Well, it would be 10 to the negative six.

The negative index tells you it's the reciprocal of the base and the exponent.

Now let's explore this a little bit further, using four to the power of five divided by four to the power of eight.

Well, we know this can be expanded as four times four times four times four times four, all divided by four times four times four times four times four times four times four times four.

Oof! So you can see why we use indices.

It's so much easier.

Now if I were to group the same number of multiplication of four, that means this would be the same using our knowledge of multiplication of fractions.

But we also know that this simplifies to one.

So that means we have one multiplied by one over or four times four times four.

And we can rewrite this in an index form as one over four to the power of three.

So therefore, we have a negative index.

One over four to the power of three is four to the power of negative three.

And we knew this anyway applying the laws of indices.

So remember the laws of indices state that when we divide and the basis are the same, we subtract our indices.

Five subtract over eight, gave us the four to the negative three.

But now it's really important to recognise that negative index tells you it's the reciprocal of the base and the exponent.

Now, what I'd like you to do is a quick check.

I want you to match the equivalents.

You've got a few equivalents here, so take your time, press pause if you need more time.

Well done.

So let's see how you got on.

Well first of all, these are our equivalents.

Three to the power of negative five is the same as one over three times three times three times three times three, which is the same as one over three to the five, which is the same as three to the power of three subtract eight, which is the same as three cubed divided by three to the power of eight.

Really well done if you got this.

Another set of equivalents is given here.

Five to the power of negative three is one over five times five times five, which is exactly the same as one over five cubed, which is the same as five to the power of five subtract eight which is the same as five to the power of five divided by five to the power of eight.

Well done.

And lastly, we have these equivalents.

Eight to the power of negative two is one over eight times eight, which is the same as one over eight squared, which is the same as eight to the power of three subtract five, which is exactly the same as eight to the power three divide by eight to the power five.

Very well done if you got these.

So given knowledge on evaluating powers, we're able to evaluate numbers with negative indices.

For example, evaluate three to the power of negative two.

We'll evaluate means to find the value of a numerical or algebraic expression.

So, given the fact that we know the negative index tells you it's the reciprocal of the base on the exponent, that means we know three to the power of negative two is exactly the same as one over three squared and therefore we can evaluate it.

We know that one over three squared is one-ninth.

So now we've evaluated three to the negative two to be one-ninth.

Now, let's have a look at another example.

We're asked to evaluate four to the power of negative three.

Now, you can see that negative index and that negative index tells you it's the reciprocal of the base on the exponent.

So that means it's one over four cubed, where we can calculate four cubed as it's 64.

So that means we've evaluated four to the power of negative three to be one over 64.

Summarising this, we now know any base to a negative power of m is exactly the same as one over a to that power of m.

So now what I'd like you to do is evaluate the following.

Take your time, press pause if you need.

Great work.

Let's see how you got on.

Well five to the power of negative three is one over five cubed, which is one over 125.

Two to the negative four is one over two to the four, which is one over 16.

Negative four, all to the power of negative three is one over the negative four or cubed, which is equal to on negative one over 64.

You may have written as one over negative 64.

They're exactly the same.

Well done if you got this right.

Laura has written this working out.

Six to the power of negative three is equal to negative one over six cubed, which is equal to negative one over 216.

This is such a common mistake.

What feedback would you give Laura? Have a little think.

Press pause for more time.

Well hopefully, you've spotted the negative index does not convert the number to a negative.

The negative index tells you it's the reciprocal of the base and the exponent.

Let's have a look at another check.

What is the value of a and b so that the answer evaluates to: a to the power b is equal to minus one over 216.

Have a little think.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well it's actually b.

The reason because if a is negative six and b is negative three, substituting in, the brackets always helps when you are using negative numbers.

Negative six, all to the power of negative three means the reciprocal of the negative six cubed, which gives me my negative one over 216.

Really well done if you got this one right.

Great work everybody.

So now it's time for your task.

So you can give it a go.

Press pause for more time.

Well done.

Let's move on to question two.

Without a calculator, evaluate the following.

Take your time, press pause if you need.

Well done.

Let's have a look at question three.

Write these numbers in ascending order.

Take your time, press pause if you need.

Fantastic work.

Question four, great question.

When a is an integer and 1<a<10, put these in ascending order.

See if you can give it a go.

Press pause as you'll need more time.

Well done.

Let's have a look at these answers.

Well, for question one you should have got these answers.

Massive well done.

Press pause if you need more time to mark.

For question two, here are our answers.

Really well done if you've got this one right, especially for e and f as you had to use your knowledge of summing fractions too.

Well done.

Press pause if you need more time to mark.

For question three, I'd evaluate each number first, giving me five, one-fifth one over 25 and one-half and then I can put them in ascending order easily.

Well done if you got this.

And for 3, b, same again, evaluating each number allows me to quickly put them in ascending order.

Really well done if you've got this.

Well for question four I'm going to rewrite a to the power of negative two as one over a squared just so it's a bit clearer for me to see and then I can put them in ascending order.

Remember, a is a value in between one and 10.

This gives me a to the power of negative two, one to the power of a, a squared, 100 a and a to the 100.

This is a great question.

A huge well done if you got this one right.

Great work everybody so now let's have a look at the exponent of zero.

Now using laws of indices, I'm going to work out the following.

Three squared times three cubed, all divided by three to the five is three to the five divided by three to the five, which is three to the zero.

Nine to the four multiplied by nine to the six, all over nine to the five multiply by nine to the five is nine to the 10 divided by nine to the 10, which is nine to the zero.

And then I've got algebraic terms, a to the b divided by a to the b is equal to a to the zero.

So what we're going to do is explore what an index of zero actually means.

Now when a number or term is being divided by itself, what is always the answer? Well, any number divided by itself always gives you one.

So, therefore we know any number or term with an index of zero is always one.

Three to the zero is one.

Nine to the zero is one, a to the zero is one.

So, I'm gonna give you this quick check.

It'll only take a few seconds and I want you to evaluate each of these.

Press pause if you need a bit more time.

Well done, well hopefully you've spotted 389 to the zero is one.

49,039 to the zero is one.

Negative three to the zero is one.

We have 1.

39302 to the power of zero is one.

Even p to the power of zero is one.

Any number with an index of zero is always one.

Now let's have a look at another check question.

Jun has given these two questions to evaluate.

Jun says they both evaluate to one.

I want you to explain if Jun is correct.

Press pause for more time.

Well done.

Let's see how you got on.

Well Jun is incorrect.

They evaluate to different numbers.

Remember to use the priority of operations.

We do know X to the zero is one.

So that means it's three multiplied by one, which gives us three.

In the second case, three to the X, well it doesn't matter what this entire number is.

Remember priority operations does state we do the brackets first.

So, this term or amount or value in our brackets is all to the power of zero, which then equates to one.

So, Jun is incorrect.

They do not both evaluate to one.

Well done.

So now what I want you to do is your task.

I want you to evaluate the following.

Please do remember the priority operations.

Take your time, press pause if you need.

Great work.

Let's have a look at the next question.

Put the following in ascending order.

I'm stating X is greater than 10.

So you can give it a go.

Press pause for more time.

Well done.

Let's move on to these answers.

For question one, you should have the following working out and answers.

Massive well done, press pause to check and to mark.

Well done.

Let's have a look at question two.

Well for question two, let's put them in order.

You should have these in ascending order.

Remember, X is greater than 10 and for b we should have these in ascending order.

Great work everybody.

That was a small learning cycle.

So, let's move on to the third part of our lesson, fractional basis.

Now a reciprocal is the multiplicative inverse of any non-zero number and any non-zero number multiplied by reciprocal is always equal to one.

Now using this definition, I want you to match the reciprocals So you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well, four and one-quarter are reciprocals.

Eight and one-eighth are reciprocals.

Two-thirds and three over two reciprocals and nine-tenths and 10 over nine are reciprocals.

Now given the definition of reciprocal, we know the negative index tells you it's the reciprocal of the base and the exponent.

So that means we know four to the negative one is equal to a quarter because one quarter is the reciprocal of four.

We know eight to the negative one is one-eighth because the reciprocal of eight is one-eighth.

That means we know two-thirds the power of negative one gives us three over two.

And this is the same as nine-tenths.

The reciprocal of nine-tenths is ten over nine.

So, the negative index tells you to use the reciprocal of the base and the exponent.

So, an index of negative one simply means the reciprocal of the number.

And this is such an important concept.

It even has its own function on a calculator.

You might be able to spot it here or sometimes seen as this.

For example, what do you think the output would be if you put into your calculator, (4/5) all to the power of negative one and you can use either one of these buttons.

What do you think the output would be? Well the output would be the reciprocal of our four-fifths, which we know to be five quarters.

Great work everybody.

So now let's move on to your task.

I want you to evaluate the following: five over two, all to the power of negative one; four over three, all to the power of negative one; two and three-fifths, that's a mixed number, all to the power of negative one; and four-fifths to the power of negative one add two to the power of negative one.

See if you can give it a go.

Press pause if you need more time.

Well done.

Let's see how you got on.

Well, the reciprocal of five over two is two over five.

We have the reciprocal of four-thirds is three over four.

Now to do the reciprocal of our mix number, two and three fifths, we had to convert it into an improper fraction first and then reciprocate it to give us five over 13.

And the last one was really good because it was a summation of two fractions.

We had the reciprocal of four-fifths is five over four and the reciprocal of two is a half.

Summing these together gives us seven over four.

Really well done if you got this.

So now we know the negative index tells you it's a reciprocal of the base and the exponent.

What do you think four-fifths, all to the power of negative two means? Have a little think.

Well, we know the negative index means we have to reciprocate.

So all I'm going to do is reciprocate, thus giving me five over four, all to the power of two.

And the index of two tells us to square the number.

So that means fiver over four multiply by fiver over four gives me an answer which can be calculated to be 25 over 16.

So therefore, the index of negative two tells us to reciprocate and then square the number.

Now what I want you to do is have a little think.

What do you think two-thirds all to the power of negative three evaluates to? Let's have a look.

Well, we know that negative index means we have to reciprocate.

So that means we end up with three over two, all to the power of three.

And then we cube our number to give me three cubed over two cubed, which I can work out to be 27 over eight.

So that means the index of negative three told us to reciprocate and then cube.

Now what I want you to do is evaluate the following.

See if you can give these a go.

Press pause for more time.

Well done.

Let's see how you got on.

Well, for a, reciprocating our three-fifths gives us five over three, all squared, which then evaluates to 25 over nine.

For b, reciprocating our three over two gives us two-thirds, then cubing gives us eight over 27.

Convert into an improper fraction first to give me nine over five.

I still have that negative two index.

So then I reciprocate and square to give 25 over 81.

And the last question, really good if you spotted this.

Convert it into a fraction first.

It's much easier.

So that means seven over 10, all to the power of negative two, reciprocate and square gives me 100 over 49.

Fantastic work if you've got this.

Excellent work everybody.

Now it's time for your task.

I want you to evaluate the following.

Press pause if you need more time.

Great work.

Let's have a look at question two.

Evaluate the following, press pause if you need more time.

Great work.

Moving on to question three, evaluate the following.

A little hint, convert to a fraction first as it will help.

See if you can give it a go.

Press pause for more time.

Fantastic work everybody.

So let's have a look at these answers.

You should have got these answers.

Press pause if you need more time to mark.

And for question two, you should have had these answers.

This is a bit of working out to help.

Press pause if you need a bit of time to mark.

Wonderful.

Let's move on to question three.

Converting it into a fraction helps, and then I've got this working out.

Press pause if you need more time to mark.

Great work everybody.

So in summary, the negative exponent tells you it's the reciprocal of the base and the exponent.

For example, four to the power of negative three is equal to one over four cubed, which is one over 64.

This also applies to fractional bases.

For example, two-thirds all to the power of negative three.

This is the same as three over two or cubed, which is the same as three cubed over two cubed, which is 27 over eight.

And sometimes, it's important to convert the fraction to an improper fraction so to evaluate.

Now, it's really important to remember that the index of negative one simply means the reciprocal of the number.

And this is such an important concept that it even has its own function on a calculator, which on some calculators is seen as this button or maybe in other calculator seen as this one.

Really, well done everybody.

It was wonderful learning with you.