exponential fun: an interactive tutorial on the properties of exponents click here to begin

exponential Fun: exponential Fun: An Interactive Tutorial on the An Interactive Tutorial on the

Properties of ExponentsProperties of Exponents

Click here to begin

objectiveobjective

to learn how to multiply and divide expressions with exponents, including zero and negative

exponents.

Tennessee State Standards:

1.0 Number and Operations1.1 demonstrate an understanding of the subsets, properties, and operations of the real

number system 1.1d use exponents to simplify a monomial written in expanded form

1.2 demonstrate an understanding of the relative size of rational and irrational numbers 1.2f select ratios and proportions to represent real-world problems (e.g. scale drawings, sampling, etc.)

2.0 Algebra2.2 use algebraic thinking to generalize a pattern by expressing the pattern in functional

notation 2.2i Evaluate an algebraic expression given values for one or more variables using grouping symbols and/or exponents less than four

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expressions containing expressions containing exponentsexponents

an expression like 46 is called a power. The exponent 6 represents the number of times the base 4 is used as a factor:

exponent

base 46 = 4 ● 4 ● 4 ● 4 ● 4 ● 4 power 6 factors of 4

multiplying exponents:multiplying exponents:product of powers propertyproduct of powers property

Let a and b be numbers and let m and n be integers:

to multiply two powers that have the same

base add the exponents together:

a2a3 = a2 ● a3 = a ● a ● a ● a ● a 2 factors 3 factors

= a3 + 2 = a5

multiplying exponents:multiplying exponents:power of a power propertypower of a power propertyLet a and b be numbers and let m and n be integers:

to find the power of a power, you multiply exponents:

(a2)3 = a2 ● a2 ● a2

3 factors

= a ● a ● a ● a ● a ● a 6 factors

= a6

= (a2)3

multiplying exponents:multiplying exponents:power of a power propertypower of a power property

to find the power of a power, you multiply exponents

Important Note: when you use the power of product property, it is the quantity within the parentheses that is raised to the power, not the individual terms. For example:

(a + 1)3 has 3 factors:(a + 1)3 = (a + 1)(a + 1)(a + 1)

multiplying exponents:multiplying exponents:power of a product propertypower of a product property


to find the power of a product, find the power of each factor and

multiply:

(a ● b)3 = a3 ● b3

(ab)3 = a3b3

zero and negative zero and negative exponentsexponents


Let a be a nonzero number and let n be a

positive integer:

A nonzero number to the zero power is 1: a0 = 1, a = 0.

a-n is the reciprocal of an: a-n = 1/an, a = 0.

dividing with exponents:dividing with exponents:quotient of powers propertyquotient of powers property


to divide powers having the same base, subtract exponents:

am/an = am – n, a = 0

dividing with exponents:dividing with exponents:power of a quotient power of a quotient

propertypropertyLet a and b be numbers and let m and n be integers:

to find a power of a quotient, find the power of the numerator and the power of the denominator and divide:

(a/b)m = am/bm, b = 0

Time for a quiz!Time for a quiz!

Now that you have learned how to multiply and divide expressions with exponents, including zero and

negative exponents, check your understanding with these quiz questions!

quiz question onequiz question oneSimplify the following expression:

(4x2y)3 ● x5

d. 64x10y3

c. 64x11y3

a. 4x10y3

b. 4x7y3

Great job!Great job!

The correct answer is c:

(4x2y)3 ● x5 = 43 ● (x2)3 ● y3 ● x5 power of a product

= 64 ● x6 ● y3●x5 power of a power

= 64x11y3 product of powers

Try again…Try again…

Remember the rules:1. To multiply two powers that have the

same base, add the exponents together.

2. To find the power of a power, you multiply exponents.

3. To find the product of a power, find the power of each factor and multiply.

quiz question twoquiz question twoSimplify the following expression:

(abc2)3(a2b)2

d. a7b5c5

c. a12b6c6

a. a5b5c5

b. a7b5c6


The correct answer is b:

(abc2)3(a2b)2 = a3b3(c2)3 ● a4b2 power of a product

= a3b3c6 ● a4b2 power of a power

= a7b5c6 product of powers




2. To find the power of a power, you multiply exponents.

3. To find the product of a power, find the power of each factor and multiply.

quiz question threequiz question threeYou are offered a job that pays 2x dollars or 2x dollars for x hours of work. Assuming you must work at least 2 hours, which method of payment would you choose?

a. 2x dollars

b. 2x dollars

Great job!Great job!The correct answer is b. If you work more than 2 hours, the pay is much better at the rate of 2x dollars per hour. For example, at

2x dollars per hour, you would earn $256 for

8 hours:

28 = 2 ● 2 ● 2 ● 2 ● 2 ● 2 ● 2 ● 2 = 256 while at 2x dollars per hour , you would earn $16:

2 ● 8 = 16


Remember the rules:an expression like 46 is called a power. The exponent 6 represents the number of times the base 4 is used as a factor:

exponentbase 46 = 4 ● 4 ● 4 ● 4 ● 4 ● 4

power 6 factors of 4

quiz question fourquiz question fourEvaluate the exponential expression:

3-2 ● 32

d. 1

c. 3

a. 0

b. 6


The correct answer is d:

3-2 ● 32 = 3-2 + 2 product of powers

= 30 add exponents

= 1 a0 is 1




2. A nonzero number to the zero power is 1: a0 = 1, a = 0.

quiz question fivequiz question fiveEvaluate the exponential expression:

(5a)-2

d. a2/5

c. 25a/2

a. 5a/2

b. 1/25a2


The correct answer is b:

(5a)-2 = 5-2 ● a-2 power of a product

= 1/52 ● 1/a2 reciprocals of 52 and a2

= 1/25a2 multiply fractions


Remember the rules:1. To find the product of a power, find

the power of each factor and multiply.

2. a-n is the reciprocal of an: a-n = 1/an, a = 0

quiz question sixquiz question sixEvaluate the exponential expression:

94 ● 92/97

d. 96/7

c. 1/9

a. 9

b. 1



94 ● 92/97 = 94+2/97

= 96/97 product of powers

= 9-1 quotient of powers

= 1/9




2. To divide powers having the same base, subtract exponents.

quiz question sevenquiz question sevenEvaluate the exponential expression:

(7/4)-3

d. 343/64

c. 64/343

a. -7/4

b. 4/7



(7/4)-3 = 7-3/4-3 power of a quotient

= 43/73 definition of negative exponents

= 64/343


Remember the rules:To find a power of a quotient, find the

power of the numerator and the power of the denominator and divide.

quiz question eight: quiz question eight: put it all together!put it all together!

Evaluate the exponential expression:

2x2y/3x ● 9xy2/y4

d. 6x/y2

c. x2/6y

a. 6x2/y

b. 18x2/y


The correct answer is a:

2x2y/3x ● 9xy2/y4 = (2x2y)(9xy2)/(3x)(y4)

= 18x3y3/3xy4 product of powers = 6x2y-1 quotient of powers

= 6x2/y definition of negative exponents


Remember the rules:1. to multiply two powers that have the

same base add the exponents together2. to divide powers having the same base,

subtract exponents3. a-n is the reciprocal of an: a-n = 1/an, a =

0

SummarySummary

Now you know how to

multiply and divide expressions with

exponents, including zero and negative

exponents.

resourcesresourcesAlgebra I Teacher's Edition. Larson, R., Boswell, L., Algebra I Teacher's Edition. Larson, R., Boswell, L.,

Kanold, T.D. and Stiff, L. 2004, Evanston, Kanold, T.D. and Stiff, L. 2004, Evanston, McDougal Littell. McDougal Littell.

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exponential fun: an interactive tutorial on the properties of exponents click here to begin

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