Lesson video

Hi, my name is Mr. Chan.

And in this lesson, we're going to look at mixed skills with negative and fractional indices.

Let's begin with this example, we're going to start off by looking at how we deal with a fraction to the power of a fraction.

So we've got four ninths to the power of three over two.

So four ninths, the power of three over two, I'm going to split the index number, which is a fraction three over two into a half multiplied by three.

So we think of this question as four nines to the power of a half inside the bracket to the power of three.

So a number to the power of a half, I know is the square root of that number so four nines to the power of half is the same as the square root of four nines.

So that's what's inside of our bracket and we've still got to cube that.

So when we're dealing with square rooting a fraction, what we can do is square root, the numerator and square root the denominator separately.

So the square root of four would be equal to the square root of nine equals three.

So we end up with two thirds inside the bracket still to be cubed.

Now two thirds cubed just means two thirds multiplied by itself three times that would equal eight over 27.

Let's have a look at another example, we've got four nines to the power of negative two.

If we think of the negative two is two multiply by negative one.

We can think of this question as four ninths inside the bracket squared to the power of negative one.

So we use the rules of indices, where we have brackets two times negative one equals negative two.

Now, when we're squaring a fraction, we just squared the numerator square, the denominator.

So we've got 16 over 81, to the power of negative one.

Now a number to the power of negative one just means we're looking for that numbers reciprocal And the reciprocal is the number.

We multiply that number by to equal one.

So what we multiply 16 over 81 by, well we flipped that fraction of a side down and that would be reciprocal because 16 over 81 multiplied by 81 over 16 would equal one.

So the reciprocal of 16 over 81 is simply 81 over 16 And that's my final answer.

Let's look at some examples where we're calculating with negative and fractional indices.

So I'm going to use the examples that we discussed previously, and I've got those results on the right hand side there, as you can see.

So in this first example, we're going to calculate four over nine to the power of three over two multiplied by four over nine to the power of negative two.

So four over nine to the power of three over two I know is equal to eight over 27 and four over nine to the power of negative two I know is a equal to 81 over 16 So my calculation for those two fractional indices becomes this calculation here.

So I would normally when I'm multiplying two fractions together, multiply the numerators and denominators separately, but I'm going to cross cancel here.

So going to cancel the eight to the 16 by divided by eight with both numbers.

So eight divided by eight gives me one and 16 divided by eight, gets me to two.

And similarly in the other diagonal, I'm going to cancel down by dividing by 27.

So 27 divided by 27 equals one and 81 divided by 27 equals three.

So I've simplified both of those fractions there.

That leaves me a quite straightforward multiplication of one multiply by three and one multiply by two, to the final answer of three over two.

Let's have a look at, If we were to divide these two fractions so four over nine to the power of three over two divided by four over nine to the power of negative two.

Again, we know the result of those two fractional indices we've discussed previously.

So we've got eight over 27 divided by 81 over 16 Now fractional division is equivalent to multiplying by the reciprocal of the second fraction.

So the calculation becomes eight over 27 multiplied by 16 over 81 Now, unfortunately I can't cross simplify here.

So I'm going to multiply the numerator and the denominator separately to get a final answer of 128 over 2,187.

Here's a question for you to try pause the video is complete the task resume the video once you're finished Here are the answers for the first question.

As you can see from the answers, I've had to work out the value for each of the cards.

So I started with each card and let's take two to the power of negative three, as an example, that would be the smallest value card that equals one eighth all the way through two there's a route 64 to the power of three that would equal 512.

So I had to find the value in order to put them into ascending order.

Here's another question for you to try pause the video to complete the task, resume the video once you finished.

Here are the answers with these types of questions.

It's important to try and work out the value of each number first before doing the calculation.

So in part , for example, two thirds to the power of negative two, I worked the answer to be nine over four, we're dividing by 27, over eight to the power of two thirds, 27 over eight to the power two thirds.

I also got as nine over four So that calculation became nine over four divided by nine over four That would equal one.

That's all for this lesson.

Thanks for watching.