Lesson video

Hi, everyone, my name is Miss 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 keywords.

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, 2 multiply by 2 multiplied by 2 is written as 2 to the power of 3.

We know the 3 is the integer exponent and identifies how many times the base has been multiplied by itself.

And the 2? Well, the 2 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 lot 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, 5 and 1/5.

These are reciprocals of one another because 5 multiplied by 1/5 is equal to 1.

2/3 and 3/2, these are reciprocals of one another because 2/3 multiply by 3/2 is equal to 1.

4 and not 0.

4 are not reciprocals of one another because 4 multiply by no 0.

4 does not give us 1.

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 bases.

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.

Well, we know 10 to the zero is 1 over 1.

We know 10 to the -1 is 1 over 10.

10 to the -2 is 1 over 100.

10 to the -3 is 1 over 1000.

And 10 to the -4 is 1 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 1 over 10 to the zero.

10 to the -1 is 1 over 10.

10 to the -2 is 1 over 10 squared.

10 to the -3 is 1 over 10 cubed.

And 10 to the -4 is 1 over 10 to the power of 4.

So therefore we have these equivalents.

10 to the -1 is 1 over 10.

10 to the -2 is 1 over 10 squared.

10 to the power of -3 is 1 over 10 cubed and 10 to the power of -4 is 1 over 10 to the 4.

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 -6.

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 4 to the power of 5 divided by 4 to the power of 8.

Well, we know this can be expanded as 4 times 4 times 4 times 4 times 4, all divided by 4 times 4 times 4 times 4 times 4 times 4 times 4 times 4.

Phew, so you can see why we use indices, it's so much easier.

Now if I were to group the same number of multiplications of 4, that means this would be the same using our knowledge of multiplications of fractions.

But we also know that this simplifies to 1.

So that means we have 1 multiplied by 1 over our 4 times 4 times 4.

And we can rewrite this in an index form as 1 over 4 to the power of 3.

So therefore we have a negative index, 1 over 4 to the power of 3 is 4 to the power of -3.

And we knew this anyway, applying the laws of indices.

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

5 subtract our 8, gave us the 4 to the -3.

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.

We've quite 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.

3 to the power of -5 is the same as 1 over 3 times 3 times 3 times 3 times 3, which is the same as 1 over 3 to the 5, which is the same as 3 to the power of 3 subtract 8, which is the same as 3 cubed divided by 3 to the power of 8.

Really well done if you got this.

Another set of equivalents is given here, 5 to the power of -3 is 1 over 5 times 5 times 5, which is exactly the same as 1 over 5 cubed, which is the same as 5 to the power of 5 subtract 8, which is the same as 5 to the power of 5 divided by 5 to the power of 8.

Well done.

And lastly, we have these equivalents.

8 to the power of -2 is 1 over 8 times 8, which is the same as 1 over 8 squared, which is the same as 8 to the power of 3 subtract 5, which is exactly the same as 8 to the power 3 divide by 8 to the power 5.

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 3 to the power of -2.

Well, 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 and the exponent, that means we know 3 to the power of -2 is exactly the same as 1 over 3 squared, and therefore we can evaluate it.

We know that 1 over 3 squared is 1/9.

So now we've evaluated 3 to the -2 to be 1/9.

Now let's have a look at another example.

We're asked to evaluate 4 to the power of -3.

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

So that means it's 1 over 4 cubed.

Well, we can calculate 4 cubed as 64.

So that means we've evaluated 4 to the power of -3 to be 1 over 64.

Summarising this, we now know any base to a negative power of m is exactly the same as 1 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 5 to the power of -3 is 1 over 5 cubed, which is 1 over 125.

2 to the -4 is 1 over 2 to the 4, which is 1 over 16.

<v ->4 all to the power of -3 is 1 over the -4, all cubed,</v> which is equal to our -1 over 64.

You may have written as 1 over -64.

They're exactly the same.

Well done, if you got this right.

Laura has written this working out.

6 to the power of -3 is equal to negative 1 over 6 cubed which is equal to negative 1/216.

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 8 of the power b is equal to minus 1/216.

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 is a is -6 and b is -3, substituting in, the brackets always helps when you are using negative numbers.

<v ->6 all to the power of -3</v> means the reciprocal of the -6 cubed, which gives me my negative 1/216.

Really well done if you've got this one right.

Great work everybody, so now it's time for your task.

So if 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, a and b are great questions.

When a is an integer and 1 is less than a, which is less than 10, put these in ascending order.

And for b, when a is a negative integer and the absolute value of a is greater than 2, put these in ascending order.

So if you can give it a go, press pause as you'll need more time.

Well done, let's all 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 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 5, 1/5, 1/25 and 1/2, and then I can put them in ascending order easily.

Well done if you got this.

And for 3b, 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 -2 as 1 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 1 and 10.

This gives me a to the power of -2, 1 to the power of a, a squared, 100a, and a to 100.

This is a great question.

A huge well done if you got this one right.

And for b, if a is a negative integer, let's see what we have.

Well same again, I'm going to convert a to the power of -2 into 1 over a squared.

This makes it a little bit easier for me to see that a squared and 1 over a squared will always give me a positive number.

And given that we know a is negative, 100 to the power of a means the reciprocal of 100 to the a, So it's 1 over 100 to the a.

Here we have an even exponent, so therefore we also know this number would be positive.

So from this information I can put the following in ascending order.

Really well done if you got this question 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.

3 square times 3 cubed, all divided by 3 to the 5 is 3 to the 5 divided by 3 to the 5, which is 3 to the zero.

9 to the 4, multiplied by 9 to the 6, all over 9 to the 5 multiplied by 9 to the 5 is 9 to the 10 divided by 9 to the 10, which is 9 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 gonna 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 1.

So therefore we know any number or term with an index of zero is always 1.

3 to the zero is 1.

9 to the zero is 1, a to the zero is 1.

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 1, 49,039 to the zero is 1, <v ->3 to the zero is 1.

</v> We have 1.

39302 to the power zero is 1.

Even p to the power zero is 1.

Any with an index of zero is always 1.

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

Jun is given these two questions to evaluate.

Jun says, "They both evaluate to 1." 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 1, so that means it's 3 multiplied by our 1, which gives us 3.

In the second case, 3 to the x, well, it doesn't matter what this entire number is.

Remember priority of 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 1.

So Jun is incorrect, they do not both evaluate to 1.

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 of 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 bases.

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

And any non-zero number multiplied by its reciprocal is always equal to 1.

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, 4 and 1/4 are reciprocals.

8 and 1/8 are reciprocals.

2/3 and 3/2 are reciprocals.

And 9/10 and 10/9 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 4 to the -1 is equal to 1/4 because 1/4 is the reciprocal of 4.

We know 8 to the -1 is 1/8 because the reciprocal of 8 is 1/8.

That means we know 2/3 to the power of -1 gives us 3/2.

And this is the same as 9/10, the reciprocal of 9/10 is 10/9.

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

So an index of -1 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, bracket, 4 over 5, close bracket, all to the power of -1? 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 4/5, which we know to be 5/4.

Great work, everybody, so now let's move on to your task.

I want you to evaluate the following: 5/2 all to the power of -1; 4/3 all to the power of -1; 2 and 3/5, that's a mixed number, all to the power of -1; and 4/5 to the power of -1 add 2 to the power of -1.

So 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 5/2 is 2/5.

We have the reciprocal of 4/3 is 3/4.

Now to do the reciprocal of our mixed number, 2 and 3/5, we had to convert it into an improper fraction first and then reciprocate it to give us 5/13.

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

We had the reciprocal of 4/5 is 5/4 and the reciprocal of 2 is 1/2.

Summing these together gives us 7/4.

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 4/5 all to the power of -2 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 5/4 all to the power of 2.

And the index of 2 tells us to square the number.

So that means 5/4 multiplied by 5/4 gives me an answer which can be calculated to be 25/16.

So therefore the index of -2 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 2/3 all to the power of -3 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 3/2, all to the power of 3.

And then we cube our number to give me 3 cubed over 2 cubed, which I can work out to be 27/8.

So that means the index of -3 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 3/5 gives us 5/3 all squared, which then evaluates to 25/9.

For b, reciprocating our 3/2 gives us 2/3, then cubing gives us 8/27.

Convert into an improper fraction first to give me 9/5, I still have that -2 index.

So then I reciprocate and square to give me 25/81.

And the last question.

Really good if you spotted this.

Convert it into a fraction first, it's much easier.

So that means 7/10, all to the power of -2, reciprocate and square gives me 100/49.

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.

So 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, 4 to the power of -3 is equal to 1 over 4 cubed, which is 1/64.

This also applies to fractional bases.

For example, 2/3 all to the power of -3.

This is the same as 3/2, all cubed, which is same as 3 cubed over 2 cubed, which is 27/8.

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 -1 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 on other calculators seen as this one.

Really well done, everybody.

It was wonderful learning with you.