WEBVTT 00:21.360 --> 00:27.695 In this section, we're going to study exponents. An exponent is a shorthand notation for repeated 00:27.760 --> 00:36.036 factors. The expression two to the fifth power is called an exponential expression. 00:36.080 --> 00:49.884 In this case, two is known as the base of the exponential expression, and five is the exponent. 00:49.920 --> 00:58.192 The exponent tells how many times the base is multiplied times itself. If 00:58.240 --> 01:08.269 x is a real number and n is a positive integer, then x to the n is the product of n factors of x. 01:08.320 --> 01:19.947 So x to the n would be x times x times x however many times over the exponent is. For two to the 01:20.000 --> 01:37.965 fifth, this is the same as two times two times two five times. This gives us thirty-two. 01:38.000 --> 01:43.404 Now we're going to evaluate some exponential expressions. Let's take a look at these 01:43.440 --> 01:51.078 first two examples. We have negative five in grouping symbols raised to the fourth power, 01:51.120 --> 01:57.718 and here we have negative five to the fourth power. There is a difference in these two 01:57.760 --> 02:05.059 problems, and that difference is the base of the exponential expression. In this first example, 02:05.120 --> 02:11.632 the base of the exponential expression is negative five. Negative five is being raised 02:11.680 --> 02:27.715 to the fourth power. So that means the factor negative five is being multiplied four times. 02:27.760 --> 02:33.954 When I multiply a negative times a negative times a negative times a negative, well 02:34.000 --> 02:43.964 that's four negatives. An even number of negatives multiplied together will give us a positive value. 02:44.000 --> 02:51.939 Five times five times five times five is six twenty-five. So this product 02:52.000 --> 02:58.846 is six hundred twenty-five. Negative five raised to the fourth power 02:58.880 --> 03:07.621 is six hundred twenty-five. Now let's take a look at the way this expression is written. 03:07.680 --> 03:14.828 Negative five to the fourth power. Notice that the negative is not in grouping symbols. 03:14.880 --> 03:22.102 It is not being raised to the fourth power. The base of this expression is five. 03:22.160 --> 03:28.842 The exponential expression is five to the fourth, and it is being multiplied by negative one. 03:28.880 --> 03:41.322 So an equivalent way of writing this is negative one times five to the fourth power. 03:41.360 --> 03:57.137 What is five to the fourth power? It is the factor of five multiplied times itself four times. 03:57.200 --> 04:03.310 We know from the previous example that five multiplied times itself four times is 04:03.360 --> 04:10.017 six hundred twenty-five, but I’m also multiplying that times negative one. 04:10.080 --> 04:19.326 So this gives us negative six hundred twenty-five. It is very important to identify what the base of 04:19.360 --> 04:28.268 your exponential expression is and what it is exactly that is being raised to the power. Let's take a look at 04:28.302 --> 04:38.045 this next example. Evaluate the expression for the given value of x. Our expression is two x to the third 04:38.080 --> 04:51.625 power, and we want to evaluate that when x is one-third. So let's substitute one-third in for x. 04:51.680 --> 04:57.564 Again, let's notice what is the base of the exponential expression that is being raised 04:57.600 --> 05:06.106 to a power. Two is not being raised to a power. Only the x is being raised to the third power. 05:06.160 --> 05:13.947 So this value that we substituted in for x, one-third, is the factor that is being repeated. 05:14.000 --> 05:25.392 We can write this as two times one-third times one-third times one-third. Notice 05:25.440 --> 05:33.801 the base of the exponential expression is the repeated factor. And so when we multiply this 05:33.840 --> 05:40.674 two is the same as two over one, and we multiply numerators across the top 05:40.720 --> 05:50.584 and then denominators across the bottom. And we get two over twenty-seven. 05:50.640 --> 05:58.592 Our next example is negative five over x squared, and we're going to evaluate this expression 05:58.640 --> 06:05.299 when x is equal to negative seven. So let's substitute our value of negative seven 06:05.360 --> 06:14.174 for the x. Notice that the x whatever it is is the base, and it's being raised to the second power. 06:14.240 --> 06:21.715 So this x of negative seven is the base. The entire thing is being raised to the second power. 06:21.760 --> 06:32.526 So we will write this as negative five divided by negative seven to the second power. 06:32.560 --> 06:42.202 We can write this portion of the exponential expression as negative seven times negative seven. 06:42.240 --> 06:58.285 The base, the factor being repeated, is negative seven, and we still have a negative and a five. Negative five in the numerator, 06:58.320 --> 07:05.792 and the denominator is negative seven times negative seven which is positive forty-nine. 07:05.840 --> 07:18.272 This gives us negative five over forty-nine. We're now going to discover some rules for operations with 07:18.305 --> 07:28.415 exponential expressions. Let's consider the expression x squared times x cubed. We're multiplying two exponential 07:28.480 --> 07:38.592 expressions with the same base. To begin let's first write this expression in expanded form. 07:38.640 --> 07:49.469 x squared times x cubed can be written as x times x. This is the base written as a factor 07:49.520 --> 08:01.315 two times, and then we're multiplying this times x cubed which can be written as x times x times x. 08:01.360 --> 08:08.188 When we're multiplying these, we notice that the parentheses really don't matter here for multiplication. 08:08.222 --> 08:21.301 And so we see that this is x times itself five times. And a shorthand way of writing this expression is x 08:21.360 --> 08:32.846 to the fifth power. So notice that x squared times x to the third is equal to x to the fifth. 08:32.880 --> 08:38.752 The product of this exponential expression with the same base times another exponential 08:38.800 --> 08:47.160 expression with the same base keeps the same base. And what's happened to our exponents? 08:47.200 --> 08:56.670 Well, two plus three is five. So we notice that when we multiply like base exponential 08:56.720 --> 09:03.543 expressions, we keep the base and add the exponents. This can be generalized 09:03.600 --> 09:14.921 as the Product Rule for Exponents. If m and n are positive integers and a is a real number, 09:14.960 --> 09:29.236 then a to the m power times a to the n power is equal to a to the m plus n. 09:29.280 --> 09:38.445 Now let's use the product rule to simplify these expressions. Six to the seventh power times six to 09:38.480 --> 09:45.719 the twelfth power. Notice that we're multiplying two exponential expressions with the same base. 09:45.760 --> 09:53.627 So our product rule tells us that the base will remain six, and then the new exponent will be 09:53.680 --> 10:08.175 the sum of the original two exponents, seven plus twelve. This gives us six to the nineteenth power. 10:08.240 --> 10:16.583 In our next example, we have two w to the ninth power times negative three w squared. 10:16.640 --> 10:24.191 We're multiplying these two expressions, and so in multiplication, we can use the commutative 10:24.240 --> 10:31.298 and the associative properties to rewrite the expression putting the like factors together. 10:31.360 --> 10:42.509 So we'll rewrite this expression as two times negative three times w to the ninth times w 10:42.560 --> 10:52.586 squared. Two times negative three is negative six. And we can use the product rule to evaluate 10:52.640 --> 11:01.061 or simplify w to the ninth times w squared. We keep the base the same, and we write the 11:01.120 --> 11:16.276 new exponent as the sum of nine plus two. This gives us negative six w to the eleventh power. 11:16.320 --> 11:23.550 Let's consider this exponential expression x squared to the third power. By the definition 11:23.600 --> 11:35.061 of exponential expressions, we can rewrite this as x squared times x squared times x squared. 11:35.120 --> 11:41.935 So we're multiplying a factor of x squared times itself three times, meaning that we can now 11:42.000 --> 11:49.943 use the Product Rule for Exponents. The bases of these exponential expressions are the same. So we 11:50.000 --> 12:03.056 keep the base of x, and we add the exponents which is the sum of two plus two plus two. 12:03.120 --> 12:15.001 Two plus two plus two can also be written as two times three because we're adding two three times. 12:15.040 --> 12:25.145 This gives us x to the sixth power. Now notice the problem that we began with had x squared raised to 12:25.200 --> 12:34.821 the third power. Can you see a shortcut method for rewriting that exponent? Instead of expanding into 12:34.880 --> 12:44.598 x squared times x squared times x squared, we can simply multiply the powers. Two times three 12:44.640 --> 12:55.542 gives us six. This leads us to the Power Rule for Exponents. If m and n are positive integers 12:55.600 --> 13:11.024 and a is a real number, then a to the m raised to the nth power is a to the m times n. 13:11.057 --> 13:17.797 Let's use the power rule to simplify these expressions. We have r to the fourth raised 13:17.840 --> 13:25.705 to the fifth power. By the power rule, we'll keep the base of the exponential expression as r, and 13:25.760 --> 13:35.949 we'll multiply these exponents, four times five, and this will give us r to the twentieth power. 13:36.000 --> 13:48.828 In our next expression, we have nine times z to the eighth to the second power. So we'll have nine, 13:48.880 --> 13:55.068 and then we'll apply the power rule to z to the eighth to the second power by keeping the base z 13:55.120 --> 14:08.748 and multiplying the powers of eight and two. And this will give us nine z to the sixteenth power. 14:08.800 --> 14:15.955 Now let's consider the expression x y to the third power. By the definition of an 14:16.000 --> 14:29.636 exponential expression, we can rewrite x y to the third power as x y times x y times x y. 14:29.680 --> 14:36.676 Now using the associative and commutative properties of multiplication, we can write this expression 14:36.720 --> 14:49.856 as x times x times x times y times y times y. We know that when we have repeated factors 14:49.920 --> 14:56.596 multiplied together, we can write them as exponential expressions. x times x times x 14:56.640 --> 15:06.606 is x to the third power, or x cubed, and y times y times y is y to the third power, 15:06.640 --> 15:20.186 or y cubed, giving us the answer x cubed y cubed as a simplified version of x y to the third power. 15:20.240 --> 15:28.261 So what can we say about evaluating or simplifying x y to the third power? 15:28.320 --> 15:37.537 Well, each of the factors inside the grouping symbols has the exponent of three applied to them. 15:37.600 --> 15:43.877 x is raised to the third power, and y is also raised to the third power. 15:43.920 --> 15:52.752 This leads us to the Power of a Product Rule. If n is a positive integer and a and b are real 15:52.800 --> 16:06.766 numbers, then a times b raised to the nth power is a to the nth power times b to the nth power. 16:06.800 --> 16:12.739 In these examples, we will use the power rule and the power of a product rule to simplify the 16:12.800 --> 16:21.080 expression. Let's begin with three x squared raised to the fourth power. We see that a 16:21.120 --> 16:28.521 product is being raised to the power of four, so we will apply the power of a product rule. 16:28.560 --> 16:37.297 We'll apply the power four to each of the factors in the grouping symbols. This gives us three to the fourth 16:37.330 --> 16:51.377 times x squared to the fourth. Three to the fourth is three times three times three times three which is eighty-one. 16:51.440 --> 16:59.085 And x squared to the fourth, we can use the power rule and rewrite this as x 16:59.120 --> 17:10.029 to the two times four, giving us eighty-one x to the eighth power. 17:10.080 --> 17:16.903 In our next example, we have negative two a b squared c to the fourth 17:16.960 --> 17:24.744 all raised to the fifth power. So again, we have a product being raised to a power, so we'll 17:24.800 --> 17:32.585 apply the power of five to each of the factors inside the grouping symbols. This gives us 17:32.640 --> 17:44.197 negative two to the fifth power times a to the fifth power times b squared to the fifth power 17:44.240 --> 17:53.640 times c to the fourth to the fifth power. And now let's simplify this expression. 17:53.680 --> 18:00.446 Negative two raised to the fifth power is taking the factor negative two and multiplying it 18:00.480 --> 18:13.626 times itself five times. This gives us negative thirty-two. Next, we have a to the fifth power. 18:13.680 --> 18:21.167 We have b squared to the fifth power. Applying the power rule, this gives us b raised to the 18:21.200 --> 18:29.943 two times five which is ten, so b to the tenth power. And here we have c to the fourth raised 18:30.000 --> 18:37.951 to the fifth. So applying the power rule will give us c to the four times five, or c to the 18:38.000 --> 18:49.562 twentieth power, giving us negative thirty-two a to the fifth b to the tenth c to the twentieth. 18:49.600 --> 18:59.205 Let's consider the expression x over y to the third power. Using the definition of an exponential expression, 18:59.238 --> 19:14.020 we can rewrite this as x over y times x over y times x over y. Now multiplying these fractions together will give us 19:14.080 --> 19:27.800 x times x times x over y times y times y. And now rewriting this using exponential notation, x times 19:27.840 --> 19:38.911 x times x is x to the third power, and y times y times y is y to the third power. 19:38.960 --> 19:46.185 So how can we generalize this rule? We see that we have a quotient raised to the third power. 19:46.240 --> 19:53.559 And in our answer here, we can see that both the numerator and denominator were raised to that power. 19:53.593 --> 20:04.003 This leads us to the Power of a Quotient Rule. If n is a positive integer and a and b are real numbers and b 20:04.037 --> 20:19.719 is not equal to zero, then a over b raised to the nth power is equal to a to the nth divided by b to the n. 20:19.760 --> 20:26.759 In these examples, we are going to use the power rule and the power of a product or quotient rule 20:26.800 --> 20:36.269 to simplify the expression. Our first expression is ten over m squared raised to the fourth power. 20:36.320 --> 20:46.345 We have a quotient being raised to the power of four. So applying that rule, we will get ten to the fourth power 20:46.379 --> 21:00.426 over m squared to the fourth power. And now evaluating ten to the fourth power in the numerator gives us ten thousand. 21:00.480 --> 21:15.007 And m squared to the fourth power gives us m to the two times four which is m to the eighth power. 21:15.041 --> 21:24.283 Our next example is three x over y to the fourth raised to the third power. We have a quotient 21:24.320 --> 21:32.358 being raised to the third power, but in the numerator, we also have a product being raised 21:32.400 --> 21:40.500 to the third power. Let's apply the quotient rule first. We can write this as three x to the 21:40.560 --> 21:49.775 third power divided by y to the fourth to the third power. And now we can apply the 21:49.840 --> 21:58.818 product rule in our numerator, giving us three to the third power times x to the third power. 21:58.880 --> 22:05.958 And in the denominator using the power rule, we have y to the fourth raised to the third 22:06.000 --> 22:12.899 which is y to the four times three, or y to the twelfth power. And then 22:12.960 --> 22:20.840 finally evaluating three to the third in our numerator which is three times three times three, 22:20.880 --> 22:36.122 this gives us twenty-seven x to the third over y to the twelfth. Let's now consider this example. x to the fifth 22:36.155 --> 22:43.029 divided by x to the third. We can apply the Fundamental Principle of Fractions and divide the numerator 22:43.062 --> 22:54.106 and denominator by a common factor. Let's write it this way first. x to the fifth can be written as x multiplied by itself 22:54.160 --> 23:04.283 five times, and x to the third can be written as x multiplied by itself three times. 23:04.320 --> 23:11.223 Now dividing out the common factors as long as x is not equal to zero would leave us 23:11.280 --> 23:25.071 with x squared. That's because x cubed is common in the numerator and the denominator. 23:25.120 --> 23:33.646 So this portion of the fraction is equal to one. Now let's see how we can generalize this. 23:33.680 --> 23:40.353 We have the common base of x, and let's look at the exponent here of two. 23:40.400 --> 23:49.795 Well, five minus three gives us two. This leads us to the Quotient Rule for Exponents. 23:49.840 --> 23:57.870 If m and n are positive integers, a is a real number, and not equal to zero, 23:57.920 --> 24:05.544 then a to the m divided by a to the n will give us a raised to the m 24:05.600 --> 24:14.520 minus n. In this section, we're going to consider quotients with exponential expressions such that 24:14.560 --> 24:24.263 m is greater than or equal to n. We'll extend this understanding in future sections. 24:24.320 --> 24:30.036 In these examples, we'll use the quotient rule to simplify the expression. 24:30.080 --> 24:36.676 We have six to the twelfth divided by six to the seventh power. So we have a quotient with 24:36.720 --> 24:45.384 exponential expressions that have the same base. We'll keep the base of six and subtract the powers 24:45.440 --> 24:57.229 twelve and seven, giving us six to the twelve minus seven or six to the fifth power. 24:57.280 --> 25:09.075 Next, we have five a to the ninth b to the fourth divided by ten a to the eighth b. Let's regroup 25:09.120 --> 25:18.184 the factors in this expression before we simplify. So we have five over ten 25:18.240 --> 25:32.198 multiplied by a to the ninth over a to the eighth times b to the fourth divided by b, which 25:32.240 --> 25:38.337 b can also be written as b to the first power. We don't want to forget that when we don't see 25:38.400 --> 25:48.347 an exponent there is an implied one as a power. Now let's simplify each of these three factors. 25:48.400 --> 26:00.826 We know that five over ten can be simplified to one over two. a to the ninth divided by a to the eighth can be rewritten as 26:00.860 --> 26:11.537 a to the nine minus eight. And b to the fourth divided by b can be written as b to the four minus one. 26:11.600 --> 26:19.879 Now taking this a step further, we have one half a nine minus eight is one. So this is a 26:19.920 --> 26:29.021 to the first power. We don't need to put the implied one as the exponent. And we have b 26:29.054 --> 26:48.340 to the third power. So we have one-half a b to the third power. This could also be written as a b to the third power over two. 26:48.400 --> 26:54.847 Now we're going to extend our understanding to discuss exponents of zero. Let's consider 26:54.880 --> 27:02.188 the expression x to the third divided by x to the third. Using the quotient rule for exponents that 27:02.240 --> 27:12.264 we recently discussed, this would give us x to the three minus three. x to the three minus three is 27:12.320 --> 27:20.506 the same as x to the zero power. Now approaching this problem from a different point of view 27:20.560 --> 27:31.083 using properties of fractions, we can rewrite this as x times x times x in the numerator and x times 27:31.120 --> 27:37.056 x times x in the denominator. And now we can divide out common factors between 27:37.120 --> 27:44.663 the numerator and the denominator. So I have a common factor of x times x times x 27:44.720 --> 27:51.003 in the numerator and the denominator. And when I divide something by itself, I get 27:51.040 --> 27:59.945 one. So we can see that by these two different approaches the value of x cubed divided by x cubed 28:00.000 --> 28:12.658 is one. Leading us to see that x to the zero is equal to one, as long as x is not zero. 28:12.720 --> 28:29.840 This tells us that any base other than zero raised to the zero power is equal to one.