WEBVTT
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In this section, we're going to study exponents. An exponent is a shorthand notation for repeated
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factors. The expression two to the fifth power is called an exponential expression.
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In this case, two is known as the base of the exponential expression, and five is the exponent.
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The exponent tells how many times the base is multiplied times itself. If
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x is a real number and n is a positive integer, then x to the n is the product of n factors of x.
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So x to the n would be x times x times x however many times over the exponent is. For two to the
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fifth, this is the same as two times two times two five times. This gives us thirty-two.
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Now we're going to evaluate some exponential expressions. Let's take a look at these
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first two examples. We have negative five in grouping symbols raised to the fourth power,
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and here we have negative five to the fourth power. There is a difference in these two
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problems, and that difference is the base of the exponential expression. In this first example,
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the base of the exponential expression is negative five. Negative five is being raised
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to the fourth power. So that means the factor negative five is being multiplied four times.
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When I multiply a negative times a negative times a negative times a negative, well
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that's four negatives. An even number of negatives multiplied together will give us a positive value.
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Five times five times five times five is six twenty-five. So this product
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is six hundred twenty-five. Negative five raised to the fourth power
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is six hundred twenty-five. Now let's take a look at the way this expression is written.
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Negative five to the fourth power. Notice that the negative is not in grouping symbols.
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It is not being raised to the fourth power. The base of this expression is five.
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The exponential expression is five to the fourth, and it is being multiplied by negative one.
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So an equivalent way of writing this is negative one times five to the fourth power.
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What is five to the fourth power? It is the factor of five multiplied times itself four times.
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We know from the previous example that five multiplied times itself four times is
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six hundred twenty-five, but I’m also multiplying that times negative one.
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So this gives us negative six hundred twenty-five. It is very important to identify what the base of
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your exponential expression is and what it is exactly that is being raised to the power. Let's take a look at
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this next example. Evaluate the expression for the given value of x. Our expression is two x to the third
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power, and we want to evaluate that when x is one-third. So let's substitute one-third in for x.
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Again, let's notice what is the base of the exponential expression that is being raised
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to a power. Two is not being raised to a power. Only the x is being raised to the third power.
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So this value that we substituted in for x, one-third, is the factor that is being repeated.
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We can write this as two times one-third times one-third times one-third. Notice
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the base of the exponential expression is the repeated factor. And so when we multiply this
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two is the same as two over one, and we multiply numerators across the top
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and then denominators across the bottom. And we get two over twenty-seven.
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Our next example is negative five over x squared, and we're going to evaluate this expression
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when x is equal to negative seven. So let's substitute our value of negative seven
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for the x. Notice that the x whatever it is is the base, and it's being raised to the second power.
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So this x of negative seven is the base. The entire thing is being raised to the second power.
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So we will write this as negative five divided by negative seven to the second power.
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We can write this portion of the exponential expression as negative seven times negative seven.
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The base, the factor being repeated, is negative seven, and we still have a negative and a five. Negative five in the numerator,
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and the denominator is negative seven times negative seven which is positive forty-nine.
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This gives us negative five over forty-nine. We're now going to discover some rules for operations with
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exponential expressions. Let's consider the expression x squared times x cubed. We're multiplying two exponential
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expressions with the same base. To begin let's first write this expression in expanded form.
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x squared times x cubed can be written as x times x. This is the base written as a factor
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two times, and then we're multiplying this times x cubed which can be written as x times x times x.
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When we're multiplying these, we notice that the parentheses really don't matter here for multiplication.
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And so we see that this is x times itself five times. And a shorthand way of writing this expression is x
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to the fifth power. So notice that x squared times x to the third is equal to x to the fifth.
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The product of this exponential expression with the same base times another exponential
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expression with the same base keeps the same base. And what's happened to our exponents?
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Well, two plus three is five. So we notice that when we multiply like base exponential
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expressions, we keep the base and add the exponents. This can be generalized
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as the Product Rule for Exponents. If m and n are positive integers and a is a real number,
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then a to the m power times a to the n power is equal to a to the m plus n.
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Now let's use the product rule to simplify these expressions. Six to the seventh power times six to
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the twelfth power. Notice that we're multiplying two exponential expressions with the same base.
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So our product rule tells us that the base will remain six, and then the new exponent will be
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the sum of the original two exponents, seven plus twelve. This gives us six to the nineteenth power.
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In our next example, we have two w to the ninth power times negative three w squared.
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We're multiplying these two expressions, and so in multiplication, we can use the commutative
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and the associative properties to rewrite the expression putting the like factors together.
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So we'll rewrite this expression as two times negative three times w to the ninth times w
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squared. Two times negative three is negative six. And we can use the product rule to evaluate
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or simplify w to the ninth times w squared. We keep the base the same, and we write the
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new exponent as the sum of nine plus two. This gives us negative six w to the eleventh power.
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Let's consider this exponential expression x squared to the third power. By the definition
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of exponential expressions, we can rewrite this as x squared times x squared times x squared.
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So we're multiplying a factor of x squared times itself three times, meaning that we can now
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use the Product Rule for Exponents. The bases of these exponential expressions are the same. So we
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keep the base of x, and we add the exponents which is the sum of two plus two plus two.
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Two plus two plus two can also be written as two times three because we're adding two three times.
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This gives us x to the sixth power. Now notice the problem that we began with had x squared raised to
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the third power. Can you see a shortcut method for rewriting that exponent? Instead of expanding into
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x squared times x squared times x squared, we can simply multiply the powers. Two times three
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gives us six. This leads us to the Power Rule for Exponents. If m and n are positive integers
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and a is a real number, then a to the m raised to the nth power is a to the m times n.
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Let's use the power rule to simplify these expressions. We have r to the fourth raised
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to the fifth power. By the power rule, we'll keep the base of the exponential expression as r, and
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we'll multiply these exponents, four times five, and this will give us r to the twentieth power.
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In our next expression, we have nine times z to the eighth to the second power. So we'll have nine,
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and then we'll apply the power rule to z to the eighth to the second power by keeping the base z
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and multiplying the powers of eight and two. And this will give us nine z to the sixteenth power.
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Now let's consider the expression x y to the third power. By the definition of an
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exponential expression, we can rewrite x y to the third power as x y times x y times x y.
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Now using the associative and commutative properties of multiplication, we can write this expression
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as x times x times x times y times y times y. We know that when we have repeated factors
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multiplied together, we can write them as exponential expressions. x times x times x
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is x to the third power, or x cubed, and y times y times y is y to the third power,
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or y cubed, giving us the answer x cubed y cubed as a simplified version of x y to the third power.
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So what can we say about evaluating or simplifying x y to the third power?
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Well, each of the factors inside the grouping symbols has the exponent of three applied to them.
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x is raised to the third power, and y is also raised to the third power.
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This leads us to the Power of a Product Rule. If n is a positive integer and a and b are real
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numbers, then a times b raised to the nth power is a to the nth power times b to the nth power.
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In these examples, we will use the power rule and the power of a product rule to simplify the
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expression. Let's begin with three x squared raised to the fourth power. We see that a
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product is being raised to the power of four, so we will apply the power of a product rule.
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We'll apply the power four to each of the factors in the grouping symbols. This gives us three to the fourth
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times x squared to the fourth. Three to the fourth is three times three times three times three which is eighty-one.
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And x squared to the fourth, we can use the power rule and rewrite this as x
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to the two times four, giving us eighty-one x to the eighth power.
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In our next example, we have negative two a b squared c to the fourth
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all raised to the fifth power. So again, we have a product being raised to a power, so we'll
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apply the power of five to each of the factors inside the grouping symbols. This gives us
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negative two to the fifth power times a to the fifth power times b squared to the fifth power
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times c to the fourth to the fifth power. And now let's simplify this expression.
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Negative two raised to the fifth power is taking the factor negative two and multiplying it
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times itself five times. This gives us negative thirty-two. Next, we have a to the fifth power.
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We have b squared to the fifth power. Applying the power rule, this gives us b raised to the
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two times five which is ten, so b to the tenth power. And here we have c to the fourth raised
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to the fifth. So applying the power rule will give us c to the four times five, or c to the
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twentieth power, giving us negative thirty-two a to the fifth b to the tenth c to the twentieth.
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Let's consider the expression x over y to the third power. Using the definition of an exponential expression,
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we can rewrite this as x over y times x over y times x over y. Now multiplying these fractions together will give us
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x times x times x over y times y times y. And now rewriting this using exponential notation, x times
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x times x is x to the third power, and y times y times y is y to the third power.
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So how can we generalize this rule? We see that we have a quotient raised to the third power.
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And in our answer here, we can see that both the numerator and denominator were raised to that power.
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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
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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.
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In these examples, we are going to use the power rule and the power of a product or quotient rule
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to simplify the expression. Our first expression is ten over m squared raised to the fourth power.
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We have a quotient being raised to the power of four. So applying that rule, we will get ten to the fourth power
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over m squared to the fourth power. And now evaluating ten to the fourth power in the numerator gives us ten thousand.
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And m squared to the fourth power gives us m to the two times four which is m to the eighth power.
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Our next example is three x over y to the fourth raised to the third power. We have a quotient
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being raised to the third power, but in the numerator, we also have a product being raised
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to the third power. Let's apply the quotient rule first. We can write this as three x to the
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third power divided by y to the fourth to the third power. And now we can apply the
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product rule in our numerator, giving us three to the third power times x to the third power.
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And in the denominator using the power rule, we have y to the fourth raised to the third
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which is y to the four times three, or y to the twelfth power. And then
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finally evaluating three to the third in our numerator which is three times three times three,
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this gives us twenty-seven x to the third over y to the twelfth. Let's now consider this example. x to the fifth
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divided by x to the third. We can apply the Fundamental Principle of Fractions and divide the numerator
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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
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five times, and x to the third can be written as x multiplied by itself three times.
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Now dividing out the common factors as long as x is not equal to zero would leave us
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with x squared. That's because x cubed is common in the numerator and the denominator.
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So this portion of the fraction is equal to one. Now let's see how we can generalize this.
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We have the common base of x, and let's look at the exponent here of two.
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Well, five minus three gives us two. This leads us to the Quotient Rule for Exponents.
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If m and n are positive integers, a is a real number, and not equal to zero,
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then a to the m divided by a to the n will give us a raised to the m
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minus n. In this section, we're going to consider quotients with exponential expressions such that
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m is greater than or equal to n. We'll extend this understanding in future sections.
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In these examples, we'll use the quotient rule to simplify the expression.
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We have six to the twelfth divided by six to the seventh power. So we have a quotient with
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exponential expressions that have the same base. We'll keep the base of six and subtract the powers
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twelve and seven, giving us six to the twelve minus seven or six to the fifth power.
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Next, we have five a to the ninth b to the fourth divided by ten a to the eighth b. Let's regroup
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the factors in this expression before we simplify. So we have five over ten
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multiplied by a to the ninth over a to the eighth times b to the fourth divided by b, which
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b can also be written as b to the first power. We don't want to forget that when we don't see
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an exponent there is an implied one as a power. Now let's simplify each of these three factors.
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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
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a to the nine minus eight. And b to the fourth divided by b can be written as b to the four minus one.
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Now taking this a step further, we have one half a nine minus eight is one. So this is a
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to the first power. We don't need to put the implied one as the exponent. And we have b
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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.
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Now we're going to extend our understanding to discuss exponents of zero. Let's consider
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the expression x to the third divided by x to the third. Using the quotient rule for exponents that
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we recently discussed, this would give us x to the three minus three. x to the three minus three is
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the same as x to the zero power. Now approaching this problem from a different point of view
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using properties of fractions, we can rewrite this as x times x times x in the numerator and x times
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x times x in the denominator. And now we can divide out common factors between
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the numerator and the denominator. So I have a common factor of x times x times x
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in the numerator and the denominator. And when I divide something by itself, I get
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one. So we can see that by these two different approaches the value of x cubed divided by x cubed
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is one. Leading us to see that x to the zero is equal to one, as long as x is not zero.
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This tells us that any base other than zero raised to the zero power is equal to one.