Lesson video

Hello, my name is Mr. Clasper and today we're going to look at the Power Law for Indices.

Let's begin with this example.

We're going to raise x to the power of two, all to the power of four.

Now, another way to write this could be x squared multiplied by x squared, multiplied by x squared, multiplied by x squared.

And using our multiplication law, we know that we can add these powers together.

So that should give us x to the power of eight.

Think about these two questions.

"What happened to the base?" and "What happened to the indices?" The base remained the same.

So the base was still x.

What happened to the indices? If we look closely, the indices have been multiplied.

So two multiplied by four would give us our eight because we had four lots of two.

Let's have a look at the general rule.

So if we have x to the power of a, all raised to the power of b, what happens to the base? And what happens to the indices? Once again, the base would remain the same.

So our base would be x.

What happens to the indices? The indices are multiplied together.

So we would have to multiply a by b, in this case.

So we have a base of x and an index of ab.

Let's try this example.

We could write this as a calculation and then apply our multiplication law.

This would give us 16x to the power of eight.

Think about these three questions.

"What happened to the base?" The base remained the same.

So the base was x in the question and the solution also had a base of x.

"What happened to the indices?" The indices were multiplied.

So multiplying four by two gave us eight as we have four lots of a power of two.

And, "What happened to the coefficient?" The coefficient in the bracket was two and we ended up with a coefficient of 16.

This is because two to the power of four would give us a value of 16.

Let's have a look at the general rule.

We're going to raise px to the power of a, all to power of b.

Let's think about our three questions again.

"What happens to the base?" The base will remain the same.

So the base in our response will be x, also.

"What happens to the indices?" We multiply the indices.

So we're going to multiply a by b, which would give us ab.

And, "What happens to the coefficient? At the moment, the coefficient is p, but we need to raise that to the power of b.

So that means our new coefficient will be p to the power of b.

So our final response would be, p to the power of b, x to the power of ab.

Here's some questions for you to try.

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

And here are your answers.

So remember, for each of the first questions from question one, we just need to apply our rule, which is that we can multiply our powers in each case.

For question 2, there are two mistakes that have been made with this.

The first one is that the coefficient should actually be eight because we need to raise two to the power of three.

And the other mistake is that the powers have been added and they should be multiplied.

So the final answer for this example would actually be 8a to the power of 15.

Let's have a look at this example.

So we have 3x to the power of a, all raised to the power of four.

And we're told this is equal to 81x to the power of 12.

Well, looking at our x to the power of 12, we know that this was made by multiplying a by four.

So we know that a multiplied by four must be equal to 12.

Therefore, a must be three.

Let's try this example.

axe to the power of three, all raised to the power of four, is equal to 16x to the power of 12.

Well, what we know in this example is that a must be raised to the power of four to get a new coefficient of 16.

So this means that a to the power of four must be equal to 16.

So something to the power of four is equal to 16.

Therefore, a must be equal to two, as two to the power of four is equal to 16.

Let's have a look at this example.

x to the power of 3a plus one, all raised to the power of four.

And we need to simplify this.

We can still apply our same rule.

So we can multiply our powers together.

So we need to multiply 3a plus one, all by four.

Another way of looking at this is just to expand this bracket.

So four multiplied by 3a plus one.

And when we expand this bracket, we should get 12a plus four.

Therefore, our solution is x to the power of 12a plus four.

Here's some more questions for you to try.

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

And here are your answers.

So let's take a look at question 4.

So we have part b.

We have pa to the power of seven, all raised to the power of three.

And we get a response of 125a to the power of 21.

So what this means is, if we look at our coefficient p, that has been raised to the power of three to get a coefficient of 125.

Therefore, that means that p would have to be five as five to the power of three gives us 125.

If you look at part c, we have ma to the power of negative three, all raised to the power of p and we get a response of 64a to the power of negative six.

Now, for this question, I actually found it easier to find the value of p first.

So if we find the value of p, that must be two as negative three multiplied by two would give us negative six.

Now that we can assume that p must be two, we can look at the coefficient inside the bracket.

So we know that m to the power of two would have to give us a coefficient of 64.

So this means that m would have to be eight as eight to the power of two would give us our value of 64.

Moving across to question 5, we can see that we have some brackets to expand.

So for the first example, we need to multiply 2y plus five, by three.

So we need to multiply all of that expression by three, or think about like an expanding bracket.

Therefore, we need to multiply 2y by three.

Sorry, I'll do that again, 'cause I stuttered.

Nearly got through it.

And here are your answers.

Let's have a look at question 4b.

So we have pa to the power of seven, all raised to the power of three, and that is equal to 125a to the power of 21.

What we need to know from this is that the coefficient p must be raised to the power of three to get a new coefficient of 125.

Therefore, that means that p must be five as five to the power of three would give us 125.

If we look at 4c, we have ma to the power of negative three, all raised to the power of p and we're told this is equal to 64a to the power of negative six.

I actually found it easier to find p first for this example.

So the value of p would have to be two, as negative three multiplied by two would give us our new power of negative six.

Now we know that the power of p is two, we know that we need to raise the coefficient m to the power of two, to get an answer of 64.

Therefore, m must be eight as eight to the power of two is 64.

If you have a look at question 5, the first example, we have a to the power of 2y plus five, all raised to the power of three.

So once again, our rule stays the same.

We multiply our two powers.

So if we multiply three by 2y plus five, we would get 6y plus 15, which is where these powers come from.

And that brings us to the end of our lesson.

So we've just learned how to apply the power law of indices.

I hope you're feeling very confident with it.

I will hopefully see you soon.