# Lesson video

In progress...

Hello, my name is Mr. Clasper.

And today we're going to look at the multiplication law for indices.

Let's begin with a recap on how to use indices.

In this example, we have three to the power of two, and three will be our base, and the number two represents our power.

How do we calculate this? Well, the power indicates how many times the base has been used in a multiplication.

So in our case, the base has been used twice, which represents a multiplication of three multiplied by three, and gives us a value of nine.

In our next example, we have a base of three, and we have a power of four.

This means that this represents a calculation with a base of three being used four times.

So three multiplied by three, multiplied by three, multiplied by three would give us a value of 81.

Let's simplify this.

So we know that three to the power of two is equal to three times by three.

And we know that three to the power of four is equivalent to a multiplication with a base of three being used four times.

Altogether, we have a multiplication with a base of three being used six times, so we get a solution of three to the power of six when we simplify.

What happened to the base and what happened to the indices? While we know that the base remained the same as it was three, and the powers, if we look carefully, have been added together.

We can apply this rule in an algebraic context.

So given this example, x to the power of two, multiplied by x to the power of four would be the same as x multiplied by x multiplied by x, multiplied by x, multiplied by x, multiplied by x.

So we've had a multiplication with a base of x being used six times, therefore this would simplify to x to the power of six.

Once again, if we think about these questions, what happened to the base? The base remained the same.

So we had x as a base for this example.

What happened to the indices? The indices were added together.

So two plus four gave us our six.

So given this example, what would happen to the base and what would happen to the indices? We know from the previous examples that the base would have to stay the same.

So the base must be x.

And the indices are always added together.

So that means in this example, this would simplify to x to the power of a plus b.

Here are some questions for you to try.

Pause the video to complete your task and click resume once you're finished.

So if we ever look at a couple of these examples, if we look at 1D, we can see we get an answer of d to the power of 13.

So we have d to the power of nine.

Remember the next one has a power of one.

And the last one has a power of three.

So adding these together would give us d to the power of 13.

If we move on to question two, we can see that the first answer is false as negative two plus negative four would give us a power of negative six.

And the last two answers are, of course, true.

Let's have a look at this example, two x squared multiplied by five x to the power of four.

This could be written like this.

This would give us an answer of 10 x to the power of six.

Questions to think about are what happened to the base? What happened to the indices? And what happened to the coefficients? Once again, our basis stayed the same, so our base was x.

What happened to the indices? The indices were still added together.

The coefficients were multiplied.

So we multiplied two by five.

Let's have a look at this example.

So we have px to the power of a multiplied by qx to the power of b.

Let's think about these three questions.

So what would happen to the base? Well, the base would stay the same.

So in our example, the base would still be x.

What happens to the indices? The indices are always added together when we multiply.

So our index would be a plus b.

And what happens to the coefficients? We multiply the coefficients.

So we would multiply p by q to get pq.

So altogether we should have pqx to the power of a plus b.

Here are some questions for you to try.

Pause the video to complete your task and click resume when you're finished.

Looking ahead at question 3C we can see we get a power of one, so c to the power of one.

This is because negative 0.

36 plus 1.

36 would give us a value of one.

And if we look at some examples from question four, we skip to 4D, we can see that we get an answer of negative 100 d to the power of 100.

So 20 multiply by negative five would give us our coefficient of negative 100 and adding our powers together would give us a total of 100.

Here are some questions for you to try.

Pause the video to complete your task, click resume once you're finished.

So the question five just make sure you're careful with negatives, particularly for 3C, so we need a value of negative six for p so that when we multiply our two coefficients together we get a new coefficient of 30.

And looking at question six, when we expand brackets, we need to make sure we multiply every term inside the bracket by any term which is outside of the bracket.

Pause the video to complete this task, and click resume once you're finished.

And here is the final solution.

So in the question, we're asked to find the area of this rectangle.

The rectangle has a length of three h to the power of five and a width of four h to the power of four.

And we should know to find the area we need to multiply the length by the width.

So when we do this, we get a final answer of 12 h to the power of nine centimetres squared.

And that brings us to the end of our lesson.

I hope you're feeling more comfortable by applying the multiplication law for indices, and I will hopefully see you soon.