WEBVTT
00:00:00.710 --> 00:00:08.680
In this video, we will learn how to simplify monomials involving single and multiple variables using the quotient rule.
00:00:09.920 --> 00:00:18.150
We will begin by recalling what we mean by exponents or indices and then define the quotient rule of exponents.
00:00:19.800 --> 00:00:26.670
Letβs firstly consider the expression four to the fifth power or four to the power of five.
00:00:28.110 --> 00:00:32.290
The four is called the base and the five is called the exponent.
00:00:32.680 --> 00:00:36.190
These are sometimes known as indices or powers.
00:00:37.500 --> 00:00:43.170
The exponent tells us the number of times that the base has been multiplied by itself.
00:00:43.970 --> 00:00:53.150
In this case, four to the power of five is equal to four multiplied by four multiplied by four multiplied by four multiplied by four.
00:00:53.560 --> 00:00:55.160
There are five fours.
00:00:56.400 --> 00:01:00.340
If we evaluate this, we get 1,024.
00:01:01.310 --> 00:01:05.540
This can also be done using a scientific calculator as shown.
00:01:07.420 --> 00:01:15.350
Often in mathematics, weβre asked to simplify expressions involving exponents but are not always required to evaluate them.
00:01:16.460 --> 00:01:22.130
This is particularly true with large exponents where the calculation would be long and cumbersome.
00:01:22.850 --> 00:01:29.980
Letβs look at how we can simplify an expression that involves a quotient of two exponential expressions.
00:01:31.150 --> 00:01:36.380
Simplify five to the sixth power divided by five cubed.
00:01:37.780 --> 00:01:43.410
We can begin this question by writing the top and bottom of the expression in expanded form.
00:01:44.360 --> 00:01:49.400
Five to the sixth power is the same as six fives multiplied together.
00:01:50.750 --> 00:01:56.760
Five cubed or five to the third power is the same as three fives multiplied together.
00:01:58.300 --> 00:02:04.210
We recall that dividing the top and bottom of a fraction by the same number does not change its value.
00:02:05.630 --> 00:02:11.280
We can therefore divide the top and bottom of this expression by five three times.
00:02:12.220 --> 00:02:17.990
Three of the fives on the numerator will cancel along with three of the fives on the denominator.
00:02:18.940 --> 00:02:28.090
This leaves us with five multiplied by five multiplied by five, which is the same as five cubed or five to the third power.
00:02:29.350 --> 00:02:33.110
We have in effect reduced each of the exponents by three.
00:02:33.590 --> 00:02:37.020
This leads us to a general rule if we have a quotient.
00:02:38.010 --> 00:02:45.220
π₯ to the power of π divided by π₯ to the power of π is equal to π₯ to the power of π minus π.
00:02:46.320 --> 00:02:54.340
When dividing or finding the quotient of two exponential terms, we can subtract the exponents or powers.
00:02:55.730 --> 00:03:04.310
This can also be written as π₯ to the power of π divided by π₯ to the power of π is equal to π₯ to the power of π minus π.
00:03:05.680 --> 00:03:09.110
This is known as the quotient rule of exponents.
00:03:10.370 --> 00:03:14.610
We will now look at some examples that we can solve using this rule.
00:03:16.080 --> 00:03:21.710
Simplify π₯ to the sixth power divided by π₯ to the fourth power.
00:03:22.810 --> 00:03:33.870
We recall that the quotient rule of exponents states that π₯ to the power of π divided by π₯ to the power of π is equal to π₯ to the power of π minus π.
00:03:35.050 --> 00:03:43.300
When finding the quotient of two exponential terms with the same base, we can subtract the exponents or powers.
00:03:44.330 --> 00:03:51.200
We can therefore work out the simplified version of our expression by subtracting four from six.
00:03:52.280 --> 00:03:58.850
This means that π₯ to the sixth power divided by π₯ to the fourth power is equal to π₯ squared.
00:04:00.940 --> 00:04:08.020
An alternative way of solving this problem if we didnβt recall the rule would be to write both terms out in full.
00:04:09.240 --> 00:04:14.120
π₯ to the sixth power is the same as six π₯βs multiplied together.
00:04:15.060 --> 00:04:21.530
In this case, we have used a dot to represent βmultipliedβ so we donβt get confused with the letter x.
00:04:22.750 --> 00:04:25.880
π₯ to the fourth power can be written in the same way.
00:04:27.200 --> 00:04:37.270
We can then divide the numerator and denominator by π₯ four times which in effect cancels four of the π₯βs on the top and bottom.
00:04:38.390 --> 00:04:43.980
Weβre left with π₯ multiplied by π₯ which, once again, is equal to π₯ squared.
00:04:45.840 --> 00:04:48.880
This method is fine if our exponents are small.
00:04:49.100 --> 00:04:54.090
However, it becomes more cumbersome if weβre dealing with larger exponents.
00:04:56.340 --> 00:04:59.150
This will be the case in our next example.
00:05:00.690 --> 00:05:06.540
Simplify four to the 17th power divided by four to the ninth power.
00:05:07.900 --> 00:05:13.850
We could begin this question by writing both the numerator and denominator out in full.
00:05:14.900 --> 00:05:19.400
The numerator would consist of 17 fours being multiplied together.
00:05:20.820 --> 00:05:23.080
This method would be very time consuming.
00:05:23.330 --> 00:05:27.500
So instead, we will use the quotient rule for exponents.
00:05:28.330 --> 00:05:36.450
This states that π₯ to the power of π divided by π₯ to the power of π is equal to π₯ to the power of π minus π.
00:05:37.520 --> 00:05:44.700
When finding the quotient of two exponential terms with the same base, we can subtract the exponents.
00:05:45.800 --> 00:05:55.970
This means that four to the 17th power divided by four to the ninth power can be rewritten as four to the power of 17 minus nine.
00:05:56.960 --> 00:06:03.450
As 17 minus nine is equal to eight, our answer becomes four to the eighth power.
00:06:06.020 --> 00:06:11.960
Our next two examples involve more complicated problems where there are more than two terms.
00:06:14.670 --> 00:06:27.830
Simplify π₯ to the 23rd power multiplied by π₯ to the 35th power divided by π₯ to the 17th power where π₯ is not equal to zero.
00:06:29.450 --> 00:06:36.740
In order to solve this problem, we need to recall two of our rules or laws of exponents or indices.
00:06:37.940 --> 00:06:40.150
Firstly, we have the product rule.
00:06:40.670 --> 00:06:48.630
This states that π₯ to the power of π multiplied by π₯ to the power of π is equal to π₯ to the power of π plus π.
00:06:49.670 --> 00:06:59.450
The quotient rule, on the other hand, states that π₯ to the power of π divided by π₯ to the power of π is equal to π₯ to the power of π minus π.
00:07:00.630 --> 00:07:09.160
When multiplying two terms with the same base, we add the exponents, whereas when dividing, we subtract the exponents.
00:07:10.230 --> 00:07:16.800
We usually begin a question of this type by simplifying the numerator and denominator first.
00:07:18.060 --> 00:07:26.000
In this question, we need to simplify π₯ to the 23rd power multiplied by π₯ to the 35th power.
00:07:27.510 --> 00:07:32.030
As the two terms are being multiplied, we need to add the exponents.
00:07:32.980 --> 00:07:36.650
23 plus 35 is equal to 58.
00:07:37.040 --> 00:07:41.690
Therefore, the numerator simplifies to π₯ to the 58th power.
00:07:43.150 --> 00:07:52.360
Our expression is therefore simplified to π₯ to the 58th power over or divided by π₯ to the 17th power.
00:07:53.740 --> 00:07:57.890
As weβre dividing here, we need to subtract the exponents.
00:07:58.710 --> 00:08:02.500
58 minus 17 is equal to 41.
00:08:03.750 --> 00:08:14.990
π₯ to the 23rd power multiplied by π₯ to the 35th power divided by π₯ to the 17th power is equal to π₯ to the 41st power.
00:08:16.230 --> 00:08:27.350
An alternative method here to keep the arithmetic as simple as possible would be to divide π₯ to the 23rd power by π₯ to the 17th power first.
00:08:28.490 --> 00:08:34.420
This wouldβve given us π₯ to the sixth power multiplied by π₯ to the 35th power.
00:08:35.860 --> 00:08:41.400
As six plus 35 is equal to 41, this wouldβve given us the same answer.
00:08:43.190 --> 00:08:47.470
Our next question is a problem involving multiple variables.
00:08:49.600 --> 00:09:02.560
Simplify π₯ to the fourth power π¦ to the fourth power multiplied by π₯ squared π¦ to the fourth power divided by π₯ to the fourth power π¦ cubed.
00:09:03.780 --> 00:09:09.360
This expression involves two variables, π₯ and π¦, which we will treat separately.
00:09:10.570 --> 00:09:14.470
We also need to recall two of our rules of exponents.
00:09:15.240 --> 00:09:17.400
These are the products in quotient rule.
00:09:17.900 --> 00:09:26.810
The product rule states that π₯ to the power of π multiplied by π₯ to the power of π is equal to π₯ to the power of π plus π.
00:09:27.820 --> 00:09:36.470
The quotient rule states that π₯ to the power of π divided by π₯ to the power of π is equal to π₯ to the power of π minus π.
00:09:37.600 --> 00:09:41.650
When multiplying two terms with the same base, we add the powers.
00:09:41.860 --> 00:09:44.280
And when dividing, we subtract them.
00:09:45.770 --> 00:09:49.120
Letβs consider our π₯-variable first.
00:09:50.590 --> 00:09:56.050
On the top or numerator, we have π₯ to the fourth power multiplied by π₯ squared.
00:09:56.500 --> 00:10:00.050
And on the denominator, we have π₯ to the fourth power.
00:10:01.450 --> 00:10:07.030
We might notice here that we have π₯ to the fourth power on the numerator and denominator.
00:10:07.350 --> 00:10:10.170
So we can divide both of these by this term.
00:10:11.220 --> 00:10:13.620
This leaves us with π₯ squared.
00:10:14.710 --> 00:10:18.460
Alternatively, we couldβve added the exponents on the numerator.
00:10:18.830 --> 00:10:21.290
Four plus two is equal to six.
00:10:21.680 --> 00:10:25.240
We couldβve then subtracted the exponents on the denominator.
00:10:25.590 --> 00:10:28.650
And six minus four is equal to two.
00:10:30.230 --> 00:10:41.420
When considering the π¦-variable, we have π¦ to the fourth power multiplied by π¦ to the fourth power divided by π¦ cubed or π¦ to the third power.
00:10:42.600 --> 00:10:50.670
Simplifying the numerator using the product rule gives us π¦ to the eighth power as four plus four is equal to eight.
00:10:51.380 --> 00:10:58.240
We can then use the quotient rule to simplify π¦ to the eighth power divided by π¦ to the third power.
00:10:59.130 --> 00:11:01.480
Eight minus three is equal to five.
00:11:01.880 --> 00:11:06.230
So our π¦-variable simplifies to π¦ to the fifth power.
00:11:07.770 --> 00:11:14.520
This means that the overall expression simplifies to π₯ squared π¦ to the fifth power.
00:11:15.800 --> 00:11:25.090
Whilst the two variables could be written in either order, we tend to follow the same format as the question which is usually alphabetical order.
00:11:28.090 --> 00:11:32.920
Our final question is a problem involving negative exponents.
00:11:35.530 --> 00:11:42.650
Simplify π₯ cubed or π₯ to the third power divided by π₯ to the sixth power.
00:11:44.310 --> 00:11:54.860
We recall that the quotient rule of exponents states that π₯ to the power of π divided by π₯ to the power of π is equal to π₯ to the power of π minus.
00:11:54.860 --> 00:11:55.240
π.
00:11:56.450 --> 00:12:00.460
If the base is the same, we can subtract the exponents.
00:12:01.540 --> 00:12:10.660
This means that, in our question, π₯ cubed divided by π₯ to the sixth power is equal to π₯ to the power of three minus six.
00:12:11.590 --> 00:12:19.930
As three minus six is equal to negative three, our expression simplifies to π₯ to the power of negative three.
00:12:21.230 --> 00:12:28.470
Letβs use another method to solve this problem to understand what a negative power or exponent means.
00:12:29.670 --> 00:12:34.700
π₯ cubed is the same as π₯ multiplied by π₯ multiplied by π₯.
00:12:35.430 --> 00:12:40.410
π₯ to the sixth power involves six π₯βs being multiplied together.
00:12:41.560 --> 00:12:47.370
This expression can be simplified by dividing the numerator and denominator by π₯.
00:12:48.390 --> 00:12:50.660
We can repeat this three times.
00:12:51.650 --> 00:13:01.550
The numerator has therefore simplified to one and the denominator to π₯ multiplied by π₯ multiplied by π₯, which is π₯ cubed.
00:13:02.850 --> 00:13:11.550
π₯ cubed divided by π₯ to the sixth power can therefore be written as one divided by or over π₯ cubed.
00:13:12.570 --> 00:13:17.600
These two results lead us to the general rule for negative exponents.
00:13:18.250 --> 00:13:27.590
This states that π₯ to the power of negative π is equal to one over π₯ to the power of π or one over π₯ to the πth power.
00:13:30.820 --> 00:13:34.300
We will now summarize the key points from this video.
00:13:36.690 --> 00:13:46.560
If you have a quotient of two exponential expressions that have the same base, then we can use the quotient rule of exponents to simplify the expression.
00:13:47.370 --> 00:13:55.070
π₯ to the power of π divided by π₯ to the power of π is equal to π₯ to the power of π minus π.
00:13:56.790 --> 00:14:01.980
This can also be written with a division sign instead of in fractional form.
00:14:03.480 --> 00:14:14.240
For example, two to the eighth power divided by two to the fifth power is equal to two to the power of eight minus five, which equals two cubed.
00:14:15.870 --> 00:14:19.520
In this video, we also used the product rule of exponents.
00:14:19.850 --> 00:14:28.040
This states that π₯ to the power of π multiplied by π₯ to the power of π is equal to π₯ to the power of π plus π.
00:14:29.270 --> 00:14:35.040
When multiplying, we add our exponents, whereas when we are dividing, we subtract them.
00:14:36.650 --> 00:14:49.190
The final question that we looked at also led us to the rule of negative exponents, which states that π₯ to the power of negative π is equal to one over π₯ to the power of π.