WEBVTT
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In this lesson, we will extend our properties of exponents to include rational exponents.
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So let's look at the definition of a to the one divided by n. If n is an integer greater than one,
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and the nth root of a is a real number, then we can say that a to the one over n
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is equal to the nth root of a. Let's try to make this a little bit easier for you to understand.
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So let's suppose that we let x equal five to the one-half power. Then x squared becomes
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five to the one half squared. And we know multiplying one half times two gives us one.
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Therefore, our answer is five to the first, or just five. So if x squared is equal to five,
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then we know that x is equal to the square root of five. Therefore, we know that the
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square root of five is equal to x, and also five to the one-half power is equal to x.
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So the square root of five is equal to five to the one-half power. Now let's look at this.
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Let's suppose that x is equal to five to the one third power. Then x cubed would equal
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five to the one third raised to the third power, which is just five. So x cubed is equal to five.
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Which means x is equal to the cube root of five. And if x is equal to the cube root of x and
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x is equal to five to the one-third, then that means the cube root of five is equal
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to five to the one-third. Now maybe what you've noticed here is that the denominator
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of this rational exponent is the index of the radical expression. The numerator of this
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rational exponent is the exponent to the radicand. Let's see if the same holds true here.
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Denominator of two index of two, numerator of one exponent of one.
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Now let's work through a few examples. We're going to rewrite each of these expressions
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using a radical, and then we're going to evaluate them. One hundred to the
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one-half power is equal to, again denominator of two index of two, which is not necessary
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to write. One hundred to the first power. Now we need to evaluate the square root of a
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hundred, which we know is the square root of ten squared, which is ten.
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Now let's look at sixty-four to the one-third power. My denominator is three. So to rewrite
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this using a radical, I'm now looking at the cube root of sixty-four to the first power.
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I can rewrite the cube root of sixty-four, as the cube root of four cubed, which is equal to four.
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Now I want you to notice the difference between this example and this last one,
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the difference being the parentheses. In this example, only x is being raised to the one-fifth power.
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However in this example, the entire expression two hundred and forty-three x
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is being raised to the one-fifth power. So because only x is being raised to the one-fifth power
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in this example, then two hundred and forty-three remains two hundred and forty-three.
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And then I can write times the fifth root of x to the first. And that's your answer two hundred
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and forty-three times the fifth root of x. Now moving on to our last example. Since both two
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hundred and forty-three and x are raised to the one-fifth power, then we can rewrite this as
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the fifth root of two hundred and forty-three times x. We can then evaluate the fifth root of two hundred
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and forty-three. Two hundred and forty-three is three to the fifth power times x. And now to
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evaluate the fifth root of three to the fifth power. I get three, and I still have the fifth root of x.
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We'll continue our study of rational exponents by looking at some rational exponents whose
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numerators are numbers other than one. Let's begin with the definition of a to the m
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divided by n. If m and n are integers greater than one with m divided by n in lowest terms,
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then a to the m divided by n power is equal to the nth root of a to the m which is equal to
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the nth root of a raised to the m power. Let's look at an example here with numbers.
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So a to the two thirds, we can rewrite a to the two-thirds as a to the one-third
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squared. Recall multiplying one-third times two in fact gives us two-thirds. Now
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to rewrite this using a radical. Our denominator is the index which is three,
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and this is a to the second power. But we could also write a to the two-thirds another way.
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a to the two-thirds could also be a squared raised to the one-third power
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again two times one third is two-thirds, and a to the two-thirds can be written as the cube root
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of a squared. So now we'll take a look at a few examples. We're going to rewrite each of
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these expressions using a radical. And then if possible, we will evaluate that radical.
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Let's begin with negative one hundred and twenty-five to the two-thirds power.
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I can rewrite negative one hundred and twenty-five to the two-thirds power as the cube root
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of negative one hundred and twenty-five to the second power. I can then rewrite that
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as the cube root of negative one hundred and twenty-five squared.
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Now we can evaluate the cube root of one of negative one hundred and twenty-five.
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I know that negative one hundred and twenty-five can be written as the cube root of negative five
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cubed. And we're squaring that. Well the cube root of negative five cubed is negative five.
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And when I square negative five, I get twenty-five. In our second example,
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we're writing the number, and we're twenty-five raising it to the three-halves power.
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To write this expression using a radical, we're going to write it as the square root of
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twenty-five cubed. We can evaluate the square root of twenty-five.
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We know that the square root of twenty-five is the square root of five squared,
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and then raised to the third power. The square root of five squared is five. And five to
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the third power is one hundred twenty-five. Our third example is n to the four-fifths.
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Rewriting this expression using a radical, I'm going to get the fifth root
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of n to the fourth power. We can't evaluate that, so this will be our final answer.
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Notice in our last example the parentheses around to around ten x. That means the expression ten x
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is raised to the two-sevenths power. To rewrite this using a radical, it's going to look like the seventh root
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of ten x squared. If we square ten x, we're going to square ten which is one hundred.
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And squaring x will give us x squared. This cannot be evaluated any further, therefore it is our final answer.
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Now that you're familiar with a to the m divided by n, let's take a look at what a
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to the negative m divided by n may look like. a to the negative m over n is equal to one
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divided by a to the positive m divided by n, as long as a to the m over n is a non-zero real number.
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So let's rewrite each of these expressions using a radical, and then we will evaluate.
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Our first one is sixteen to the negative one-fourth power. I'm going to rewrite this
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as one divided by sixteen to the positive one-fourth power. That's our definition of a
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to the negative m divided by n. Now let's focus on writing sixteen to the one-fourth power
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using a radical. This becomes one divided by the fourth root of sixteen to the first power.
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I know that sixteen can be written as two to the fourth power. And then to finish evaluating
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this, I know that the fourth root of two to the fourth is just two. So our answer is one half.
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Now in our second example, we have nine to the negative five halves power.
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Again we're going to use our definition of a to the negative m over n
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to rewrite this as one divided by nine to the positive five halves power.
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Now we'll focus on rewriting nine to the five halves using a radical.
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This will look like the square root of nine to the fifth power. I know that the square root of nine is three.
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So one divided by three to the fifth which is equal to one divided by two hundred and forty-three.
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And that's the answer to that problem. Now we're going to use our rules for exponents
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to simplify expressions. Here is a summary of exponent rules, all of which we have studied before.
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In our first example, we're going to multiply, and we want to assume that all variables
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represent positive real numbers. So here we're taking b to the one third, and we're multiplying
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it times b to the eleven-thirds. So here we are multiplying like bases, and so we know
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that we're going to add our exponents. We can write this as b to the one-third plus
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eleven-thirds, which we know is b to the twelve thirds. And twelve divided by three is four.
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So this is equal to b to the fourth power. Now in our second example, what we're doing here is
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we're multiplying x to the one-fourth times x to the eighth minus three x. So again, we're going
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to be adding our exponents. So x to the one-fourth times x to the one, I'm sorry,
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x to the eighth is going to give us x to the one-fourth plus eight minus,
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now we're taking x to the one-fourth and we're multiplying it times
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three x. That's going to give us three x. Now remember this is x to the first power.
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There is an understood one as the exponent to x here. So x to the one-fourth times x to the first
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is going to give us x to the one-fourth plus one. Now we can add these numbers
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together in the exponents position. And that's going to give us one-fourth plus
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eight. Gives us x to the thirty-three fourths. Minus three x one-fourth plus one gives us
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five-fourths, and this is our final answer to this example.
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Let's look at our next example, we're going to use rational exponents to write each of
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these as a single radical expression. And we are to assume that all variables represent
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positive real numbers. So in my first example I have the cube root of t
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times the fourth root of t. So first we're going to rewrite each of these using
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rational exponents. I can write the cube root of t, now remember this is understood t
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to the first power. So the cube root of t, I can write as t to the one third. Again the
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numerator is the exponent here, and the denominator is your index. Now to write
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the fourth root of t using a rational exponent. That's going to look like t to the one-fourth.
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And now what we have here is t to the one third times t to the one-fourth, which
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remember multiplying like bases, we're going to add the exponents. This will give
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me t to the one-third plus one-fourth. And that is equal to t to the seven twelfths.
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Now let's take t to the seven-twelfths and rewrite that using a radical.
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Your denominator is the index, so this will look like the twelfth root of t
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to the seventh power. Now in our next example, you'll notice we are dividing.
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But first let's rewrite each of these radical expressions using a rational exponent.
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The cube root of t, I can write as t to the one-third. Divided by the fourth root of t,
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looks like t to the one fourth. Now in this case we are dividing like bases. And when we
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divide like bases, we subtract the exponents. Subtracting my exponents will give me t
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to the one-third minus one-fourth. t to the one-third minus one-fourth is t to the one-twelfth.
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And now we will take t to the one-twelfth, and re-write is using a radical. That will
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look like the twelfth root of t to the first power which is just the twelfth root of t.
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Let's continue our practice of working with rational exponents. We're going to use the properties
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of exponents to simplify each of these expressions. We're going to write them with
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positive exponents. And we are to assume that all variables represent positive real
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numbers. So in our first example, we have four u to the negative two raised to the three-
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halves power. So recall that when we have a base raised to an exponent and raised to
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another exponent, we're multiplying those exponents. So we can rewrite this as four to
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the three-halves power times u to the negative two times three-halves. Four to the
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three-halves power, we can write using a radical as the square root of four
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to the third power times negative two times three-halves will give us negative three. So u
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to the negative two times three-halves is the same as u to the negative three. Now we
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know that the square root of four can be written as two squared. And since our
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directions say to write using positive exponents. Then I can rewrite u to the
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negative third power, as one divided by u to the positive third power. Now the square
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root of two squared is just two. And two cubed is eight. Eight times one divided by
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u cubed will give us eight divided by u cubed as our final answer.
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Looking at our second example, I notice in my numerator we're multiplying like bases,
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which means we're going to add our exponents. Let's do that first. m to the three-fifths
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times m to the one-fifth will look like m to the three-fifths plus one-fifth. And we're
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dividing that by m to the negative one fifth. Three-fifths plus one-fifth is four-fifths.
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So m to the three-fifths plus one-fifth is m to the four-fifths. And we are
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dividing that by m to the negative one-fifth. Remember when we divide like bases,
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we subtract our exponents. So m to the four-fifths divided by m to the negative one-fifth
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can be written as m to the four-fifths minus negative one-fifth. And we know that
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subtracting negative one-fifth is the same as adding one-fifth, so this is equal to m to
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the four-fifths plus one-fifth, which is m to the five-fifths. And five divided by five is one.
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So this is equal to m to the first power, or just m.