Lesson video

Hi everyone, my name is Ms. Cou, and today, we're looking at Arithmetic Procedures, Index Laws.

I really hope you enjoy the lesson.

It's going to be challenging.

It's got lots of fun tasks to do.

I know I'm going to enjoy teaching it.

So let's make a start.

Hi everyone, and welcome to this lesson on the laws of indices, fractional exponents, and it's under the unit, Arithmetic Procedures, Index Laws.

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

Now let's recap on some keywords.

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 1/5.

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

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

A non-example would be four and 0.

4.

These are not reciprocals of each other because four multiplied by 0.

4 does not give you one.

Today's lesson will be broken into three parts.

We'll be looking at unit fraction exponents first, then fractional exponents, and then problems involving fractional exponents.

So let's make a start looking at unit fraction exponents.

Now using the laws of indices, I want you to quickly work out the answers to this.

Press pause if you need a bit of time.

Well done.

Well, hopefully you should've got nine.

Nine to the power of one is fine.

100, 100 to power of one is fine.

And 49, 49 to the power of one is also fine.

Now, I'm going to look at the first calculation.

Nine to the 1/2 times nine to the 1/2 is equal to nine.

Now given the fact that we know nine to the 1/2 is exactly the same number as nine to the 1/2, we basically have a number multiplied by itself that gives nine.

So what I want you to do is think about, what number multiplied by itself gives nine? Well, it's got to be three, so that means we know, nine to the power of 1/2 has to be three.

So that means we have the calculation, three multiplied by three is equal to nine.

I'm gonna show you the next one.

100 to the 1/2 multiplied by 100 to the power of 1/2 is equal to 100.

Now we know 100 to the 1/2 is exactly the same as 100 to the 1/2.

So what number multiplied by itself gives 100? Well, that means we know 100 to the 1/2 has to be 10, because 10 multiplied by 10 is 100.

Let's have a look at this last one.

49 to the 1/2 times 49 to the 1/2 we know is 49.

So we're asking ourselves the same question.

If 49 to the 1/2 is the same as 49 to the 1/2, what number multiplied by itself gives 49? Well, it's gotta be seven.

So that means 49 to the 1/2 has to be seven because seven times seven is 49.

So let's summarise this.

Nine to the 1/2 is equal to three, 100 to the 1/2 is equal to 10, and 49 to the 1/2 is equal to seven.

Can you identify what the index of 1/2 actually means? Have a little think.

Well, it means the square root, because the square root of nine is three, the square root of 100 is 10, and the square root of 49 is seven.

So the exponent 1/2 means the square root of the base.

What I want you to do now is I'm going to check that understanding a little more.

I want you to evaluate the following.

Take your time and press pause if you need.

Well done, let's see how you got on.

Remember we know the exponent of 1/2 means the square root of the base.

So that means it's the square root of 25, which is five.

For b, 36 to the 1/2 means the square root is 36 which is six.

For c, we have the square root of 121, which is 11.

For d, the square root of 64 is eight, then we have 64 to the zero, which is one, so it's eight subtract one which is seven.

49 to the 1/2 is the same as the square root of 49.

Subtract the square root of 36, which is six, so that means it's seven subtract six, which is one.

f, well, the square root 144 is 12, over the square root of four is two, giving me six.

Really well done if you got this.

So now we're going to look at the exponent of 1/3.

Same again, I want you to use laws of indices and work out the following answers.

Press pause if you need more time.

Well done, let's see how you got on.

Well, it's eight or eight to the one, 27 or 27 to the power of one, and 1,000 or 1,000 to the power of one.

So let's look at this a little bit more.

Go to look at the first calculation, eight to the power of 1/3 times eight to the power of 1/3 times eight to the power of 1/3 of eight.

What number does eight to the power of 1/3 represent to make this calculation correct? Have a little think.

Well, it has to be two, because two multiplied by two multiplied by two is eight.

Let's have a look at the next one.

27 to the 1/3 times 27 to the 1/3 times 27 to the power of 1/3 is 27.

So what number do you think 27 to the power of 1/3 represents to make this calculation correct? Well, it has to be three.

So that means 27 to the power of 1/3 has to be three because it gives us the calculation of three times three times three, which is 27.

Lastly, let's have a look at 1,000 to the power of 1/3, multiplied by 1,000 to the power of 1/3 multiplied by 1,000 to the power of 1/3.

That makes 1,000.

So what number does 1,000 to the power of 1/3 represent to make this calculation correct? Well, it has to be 10, because we know 10 times 10 times 10 is 1,000.

Now let's summarise.

Eight to the power of 1/3 is two.

27 to the power of 1/3 is three.

And 1,000 to the power of 1/3 is 10.

What do you think the index of 1/3 represents? Well, hopefully you spotted the index of 1/3 represents the cube root.

The cube root of eight is two, the cube root of 27 is three, and the cube root of 1,000 is 10.

So the exponent of 1/3 means the cube root of the base.

Another way to understand the meaning of the fractional exponent is to use equations.

For example, a to the power of m multiplied by a to the power of m gives us a to the power of one.

So if I were to rewrite this using my laws of indices, this is the same as a of the power of m all squared, which is equal to a.

Then, using our inverse, the inverse of squared is square root so that means a to the power of m is equal to the square root of a.

Now given the fact that we know a to the m times a to the m is equal to a to the one, that means writing an equation using the exponents, 2m must equate to one.

So therefore, m is 1/2.

So that means we know a to the 1/2 is equal to the square root of a.

Now using these approaches, what do you think the following fractional indices represent? Have a little think.

Press pause if you need more time.

Now, a to the power of 1/4 means the fourth root of a.

a to the power of 1/5 means the fifth root of a.

And a to the power of 1/9 means the ninth root of a.

Well done if you got this.

Now what I want you to do is a quick check.

I want you to match the following from left to right.

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, 64 to the 1/2 means the square root of 64.

64 to the 1/3 means the cube root of 64.

16 to the 1/4 means the fourth root of 16, and 16 to the 1/2 means the square root of 16.

Let's have a look at what they evaluate to.

Well, the cube root of 64 is four.

The square root of 64 is eight.

The square root of 16 is four, and the fourth root of 16 is two.

Well done.

So let's summarise what we know.

a to the power of 1/m is equal to the mth root of a.

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

Which of the following evaluate to an integer? See if you can give it a go.

Press pause if you need more time.

Let's have a look.

Well, 64 to the power of 1/2 means the square root of 64 which is eight, and this evaluates to an integer.

64 to the power of 1/3 means the cube root of 64, which is four.

Eight to the power of 1/2 means the square root of eight.

Eight is not a perfect square number.

So that means it does not evaluate to an integer.

Eight to the power of 1/3 means the cube root of eight, which is two, and that does evaluate to an integer.

Really well done if you got this.

Great work everybody.

Now it's time for your task.

I want you to evaluate the following, take your time, show working out where you need.

Great work.

Let's look at question 2.

Question 2 is a nice little task.

When each number evaluates to a number, shade in that square, and the shaded square reveals a path.

You simply need to identify which letter is the start of the path, which letter is the end of the path, and then I want you to sum all the integers that make that path.

So you can give it a go.

Press pause if you need more time.

Well done, let's have a look at these answers.

Well for question 1, here's all the working out and these are the answers.

Press pause if you need more time to mark.

Great work, let's look at question 2.

Well, if I'm shading in all those numbers which evaluate to an integer, we should have this.

And then, identifying the start and the end, we should have this path, summing up all of those integers gives us 52.

So we start at C, finish at D, and the sum of the integers sum to 52.

Well done if you got this.

Great work everybody.

So now let's have a look at fractional exponents.

We're going to look at fractional exponents with a numerator greater than one, for example, 16 to the power of 3/2.

Now using the laws of indices, we know this is the same as 16 to the power of 1/2 all cubed.

Now remember the laws of indices state, when raising the power, we can multiply the exponents.

And we know these are equivalent because 1/2 multiplied by three is exactly the same as 3/2.

Now using a unit fraction first and then raising to the power does make the calculation much easier.

For example, let's look at 16 to the power of 3/2.

Writing using a unit fraction, this is the same as 16 all to the power of 1/2, and then all cubed.

Then knowing 16 to the power of 1/2 means the square root of 16, which I then need to cube, I can work the square root to 16 out to be four and then cube it to give me a final answer of 64.

So that means 16 to the power of 3/2 is simply 64.

Now, I'm going to do a check.

I'm going to do the first one, and I'd like you to do the second one.

We're going to evaluate eight to the power of 2/3.

Well, let's rewrite this using a unit fraction first.

This is the same as eight to the power of 1/3, all squared.

I know eight to the power of 1/3 means the cube root of eight, which I then am going to square.

The cube root of eight is two, squared then gives me four.

So eight to the power of 2/3 is equal to four.

Next, I want you to do a quick check.

Evaluate 25 to the power of 3/2.

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, first of all, rewriting as a unit fraction does help.

So this is the same as 25 to the power of 1/2, all cubed.

This then is the same as the square root of 25 all cubed, which is five cubed, which I can evaluate to 125.

So that means 25 to the power of 3/2 is 125.

Well done if you got this one right.

So in summary, what do you think a the power of m/n evaluates to? This would be the nth root of a, all to the power of m.

Izzy also says, "It can also summarise to," the nth root of a to the power of m.

Do you think Izzy's correct? And I want you to explain.

Well yes, they are both correct.

If you notice, using our laws of indices, they both equate to exactly the same thing.

The first approach finds the root first, and then applies the exponent.

The second approach applies the exponent first, and then applies the root.

So what I'm going to do now is show you a quick check.

This is the application of those two different approaches.

Both Izzy and Laura do the same calculation.

Which one do you prefer, and why? Press pause for more time.

Well, both methods do give the correct answer.

However, without a calculator, Laura's method is much easier to work out.

In other words, finding the fourth root of 81 first, which is three, and then cubing to give 27.

Izzy's method wanted you to cube 81 first, which was a huge 531,441, and then find the fourth root of that, which is 27.

So definitely without a calculator, Izzy's method is a lot harder.

Great work everybody.

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

Take your time, show you're working out as well.

Press pause if you need.

Fantastic everybody, so let's move on to question 2.

Write the correct exponent to make the number in the centre.

See if you can give it a go.

Press pause if we need more time.

Well done.

Let's see how you got on.

Question 1, we should have had all of these answers.

Here's a bit of working out, press pause if you need to check and mark.

For the continuation of question 1, here are all our workings out and our answers.

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

Well done, let's have a look at question 2.

This was a really tough question.

Superb if you got this one right.

Great work everybody.

So now let's have a look at the last part of our lesson, problems involving fractional exponents.

So sometimes, the answer doesn't always give an integer, and as such, we must use our knowledge on the laws of indices indicated here.

So using the laws of multiplication when the bases are the same, we add our exponents.

Using the law of division, when the bases are the same, we subtract our exponents.

When raising the power, we multiply your exponents.

When you have a to the power of 1/m, it means the mth root of a, and when you have a to the power of m/n, it means the nth root of a all to the power of m.

We have lots of lovely laws of indices here.

So what we're gonna do is we're going to work out the exact value of five to the power of 7/2 using our knowledge on laws of indices.

Well first of all, we're going to write a unit fraction to a raised power first.

This means five to the 1/2, all to the power of seven.

And then I'm just gonna expand this.

It means we have seven lots of five to the 1/2, all multiplied by each other.

And now what I'm gonna do is use our laws of indices.

We know five to the 1/2 multiplied by five to the 1/2 is five, or five to the one.

We know this five to the 1/2 multiplied by this five to the 1/2 also gives us five to the one.

And we know this five to the 1/2 multiplied by this five to the 1/2 gives us five to the one, and then we simply have our five to the 1/2 left over.

Well I can work out these multiplications of five to the one.

Five times five times five is 125.

And then I'm simply multiplying it by, well I do know that exponent of 1/2 means the square root, so that means I'm multiplying it by the square root of five.

Simplifying my answer, we have now worked out the exact value of five to the power of 7/2, and that's 125 root five.

Now it's time for your check.

I want you to work out two to the power of 9/2 as an exact value.

See if you can give it a go.

Press pause for more time.

Well done, let's see how you got on.

Well first of all, let's write it using our unit fraction.

Two to the power of 1/2 all to the power of nine.

What does this mean? Well, it means we have nine lots of two to the power of 1/2, all multiplied by each other.

Then I'm just gonna group them, because I know two to the 1/2 times two to the 1/2 is two to the one.

I have four lots of two to the one or four lots of multiplications of two, and then I have a multiplication of two to the 1/2.

Well if I work out the two times two times two times two, that gives me 16, and I know two to the power of 1/2 is exactly the same as the square root of two.

I have now written my exact value of two to the power of 9/2, and it's 16 root two.

Great work everybody.

Now it's time for your task.

What I want you to do is write the following as an exact answer.

See if you can give it a go.

Press pause for more time.

Fantastic.

Let's move onto these answers.

You should have got the answer for part a to be 100 root 10.

And the answer for part b should have been 27 multiplied by the cube root of three.

That was a great question.

Amazing work, everybody.

It was tough today, and in this lesson, we've recognised the unit fraction exponent represents the root of the base of the number.

For example, 25 to the power of 1/2 represents the square root of 25.

And we've summarised it as a to the power of 1/m is equal to the mth root of a.

Now when the fractional exponent has a numerator greater than one, we are applying multiple laws of indices.

For example, 64 to the power of 2/3 is the same as the cube root of 64 all squared, which evaluates the 16.

And we summarise this as a to the power of m/n is equal to the nth root of a, all to the m.

Really, really well done everybody.

It was tough lesson today.

Fantastic learning with you.