Properties of Exponents



Core Plus Mathematics IV Name_____________________

Properties of Exponents Worksheet Per. _______

Properties of Exponents

Definition of Natural Number Exponents:

Factor Cancellation Property of Fractions: [pic]

Fraction Multiplication: [pic]

Product Rule of Exponents

|[pic] |= | |Use the definition of exponents to rewrite the expression as a product |

| | | |without exponents |

| |= | |Now use the definition of exponents to rewrite the product as an |

| | | |exponential expression (an expression with just a base and one exponent) |

|[pic] |= | |Rewrite this expression as an exponential expression with just one |

| | | |exponent |

|[pic] |= | |Generalize your result and rewrite this expression as single exponential |

| | | |expression |

Power Rule of Exponents

|[pic] |= | |Use the definition of exponents to rewrite the expression inside of |

| | | |parenthesis as a product without exponents |

| |= | |Use the definition of exponents to rewrite the previous expression without|

| | | |exponents |

| |= | |Now use the definition of exponents to rewrite the product as single |

| | | |exponential expression (an expression with just a base and one exponent) |

|[pic] |= | |Rewrite this expression as single exponential expression |

|[pic] |= | |Generalize your result and rewrite this expression as single exponential |

| | | |expression |

Power of a Product Rule

|[pic] |= | |Use the definition of exponents to rewrite the expression as a product |

| | | |without exponents |

| |= | |Now use the commutative and associative properties of multiplication to |

| | | |rewrite this product so that the a factors are together and the b factors |

| | | |are together |

| |= | |Rewrite the expression so that the factor a is raised to an exponent and |

| | | |the factor b is raised to an exponent |

|[pic] |= | |Rewrite the expression as a product of factors raised to exponents |

|[pic] |= | |Generalize your result and rewrite the expression as a product of factors |

| | | |raised to exponents. |

Power of a Quotient Rule

|[pic] |= | |Use the definition of exponents to rewrite this expression as a product |

| | | |without exponents. |

| |= | |Now use the definition of fraction multiplication to rewrite the |

| | | |expression as a single fraction without exponents |

| |= | |Rewrite the expression as single fraction with an exponential expression |

| | | |in the numerator and an exponential expression in the denominator |

|[pic] |= | |Rewrite the expression as single fraction with an exponential expression |

| | | |in the numerator and an exponential expression in the denominator |

|[pic] |= | |Generalize your result and rewrite the expression as single fraction with |

| | | |an exponential expression in the numerator and an exponential expression |

| | | |in the denominator. |

Quotient Rule

|[pic] |= | |Use the definition of exponents to rewrite this expression as a single |

| | | |fraction without exponents in the numerator or denominator |

| |= | |Use the factor cancellation property of fractions to reduce this fraction.|

| |= | |Rewrite the expression as a single exponential expression (no fractions) |

|[pic] |= | |Rewrite the expression as a single exponential expression (no fractions) |

|[pic] |= | |Generalize your result and rewrite the expression as a single exponential |

| | | |expression (no fractions) |

Extending the Definition of Exponents

We have defined exponents according to this definition

But using this definition restricts us to the natural numbers = {1, 2, 3, ….}. After all, what would it mean to have zero factors of x? Or – 2 factors of x? Or ¾ factors of x?

Whenever mathematicians define a symbol or concept on a subset of the real numbers, they like to extend it to bigger sets of the real numbers, and if possible to the whole real number system.

So let’s try to extend the definition of exponents to the whole numbers = {0, 1, 2, 3, ….}. The only number in the whole numbers that is not in the natural numbers is the number zero.

When mathematicians extend the definition of a concept they do so that all the properties that were true on the smaller set of numbers are also true on the larger set of numbers. We’ll extend the definition of exponents to include zero in such a way that the quotient rule is still true.

Exponent of Zero

|[pic] |= | |Rewrite the numerator and denominator of the fraction as a whole number |

| | | |using the definition of exponents |

| |= | |By definition of fractions, what whole number does the previous fraction |

| | | |equal. |

|[pic] |= | |In general what whole number does this fraction equal (if [pic]) |

|[pic] |= | |Now use the quotient rule from above to write this expression as a single |

| | | |base that’s raised to an exponent that is the difference of two values |

| |= | |Rewrite the expression as a base raised to a whole number |

| | | | |

|[pic] |= | |If two expressions are equal to the same thing, then they must equal each |

| | | |other. Given the two expressions above that are equal to [pic], what can |

| | | |we conclude about [pic]? |

Negative Integer Exponents

Now that we have extended the definition of exponents to whole numbers, let’s see if we can extend the definition to the integers = {…-2, -1, 0, 1, 2, 3, …}. Of course the integers contain the whole numbers, so we just have to extend the definition to the negative integers.

Again we will extend the definition in such a way that the quotient rule is still true.

|[pic] |= | |Rewrite the numerator and denominator of the fraction as products using |

| | | |the definition of exponents |

| |= | |Use the cancellation property of fractions to reduce the fraction. |

|[pic] |= | |Now use the quotient rule to rewrite this expression as the base raised to|

| | | |the difference of two numbers |

| |= | |Rewrite the expression as a base raised to an integer. |

| | | | |

| |= | |We have two expressions equal to [pic], therefore they must equal each |

| | | |other. Set these two expressions equal to each other. |

|[pic] |= | |Generalize your result to [pic] |

| | | | |

| | | | |

Rational Number Exponents

Now let’s extend the definition to rational numbers. The rational numbers remember are numbers that can be written as p/q where p and q are both integers. All integers are also rational numbers since an integer can always be expressed as a fraction with a denominator of one (eg. 7 = 7/1). So we only have to extend the definition to rational numbers that are not integers such as 2/3.

But before we do this, we will extend the definition just to rational numbers that have a numerator of one, such as 1/3 or 1/5.

|[pic] |= | |Use the product rule to rewrite this expression as a base raised to the |

| | | |sum of 3 numbers |

| |= | |Now rewrite the expression as a base raised to one number |

| |= | |Now rewrite the expression as a whole number not raised to any base. |

Rational Number Exponents (Continued)

|[pic] |= | |So [pic] is a number that when I multiply it by itself three times equals |

| | | |2. Can you think of another notation to use to represent a number that |

| | | |when we multiply it by itself three times equals two? Think how we would |

| | | |represent a number that when multiplied by itself equals 2. |

|[pic] |= | |Generalize your result and define an exponent of the form 1/n. |

| | | | |

| | | | |

Now we want to define exponents for any type of rational number, not just those that have a numerator equal to one.

|[pic] |= | |Rewrite the expression as an exponential expression with an exponent that |

| | | |is the product of two fractions (Hint: One fraction should have a |

| | | |numerator of one and the other fraction should have a denominator of one) |

| |= | |Use the power rule of exponents to rewrite the expression as an |

| | | |exponential expression in parenthesis and an exponent outside of |

| | | |parenthesis such that the exponent inside parenthesis has a numerator of |

| | | |1. |

| |= | |Now rewrite the expression inside parenthesis in radical notation |

|[pic] |= | |Generalize your result and define an exponent of the form m/n. |

| | | | |

| | | | |

Evaluate [pic] without a calculator using the definition above.

|[pic] |= | |Rewrite the expression as an exponential expression with an exponent that |

| | | |is the product of two fractions |

| | | | |

| |= | |Use the power rule of exponents to rewrite the expression as an |

| | | |exponential expression in parenthesis and an exponent outside of |

| | | |parenthesis with a denominator of one |

| |= | |Now rewrite the expression inside parenthesis in radical notation |

| | | | |

| |= | |Evaluate the radical inside parenthesis |

| | | | |

| |= | |Raise the number inside parenthesis to the exponent outside the |

| | | |parenthesis |

We now know how to evaluate exponents for any rational number.

|Question: |Answer: (Use a complete sentence) |

| | |

|For what type of real | |

|number do we not know how | |

|to evaluate exponents? | |

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Special Fraction Rule

|[pic] |= | |Rewrite the expression using the rule for negative integer exponents |

| |= | |Use the exponent rule for fractions to rewrite the denominator of the |

| | | |previous expression as fraction with an exponential expression in the |

| | | |denominator and the numerator. |

| |= | |Rewrite the expression as the division of two fractions using a division |

| | |[pic] |symbol |

| | | |Divide the two fractions according to the rule for fraction division |

| | | | |

| | | | |

| | | |Rewrite the expression as a fraction raised to an exponent using the |

| | | |fraction property of exponents |

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n factors of a

I’m the exponent. I’m up high and smaller

I’m the base

Together we form an exponential expression

[pic]

[pic]

n factors of x

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