8.1.1B Integer Exponents & Scientific Notation

Number & Operation
Standard 8.1.1

Read, write, compare, classify and represent real numbers, and use them to solve problems in various contexts.

Benchmark: Integer Exponents

Know and apply the properties of positive and negative integer exponents to generate equivalent numerical expressions.

For example: [math]3^{2}\times 3^{\left ( -5 \right )}=3^{\left ( -3 \right )}=\left ( \frac{1}{3} \right )^{3}=\frac{1}{27}[/math].

Benchmark: Scientific Notation & Significant Digits

Express approximations of very large and very small numbers using scientific notation; understand how calculators display numbers in scientific notation. Multiply and divide numbers expressed in scientific notation, express the answer in scientific notation, using the correct number of significant digits when physical measurements are involved.


For example: (4.2×104) × (8.25×103) = 3.465×108, but if these numbers represent physical measurements, the answer should be expressed as 3.5×108 because the first factor, 4.2×104, only has two significant digits.


Big Ideas and Essential Understandings 
Standard 8.1.1 Essential Understandings

The focus of instruction in this standard is on understanding the real number system. Students have only experienced rational numbers in the form of whole numbers, integers, decimals and fractions. Students will expand their understanding of rational numbers as they represent and operate with very large or very small numbers in scientific notation and exponential form. Students complete their understanding of the real number system as they are introduced to irrational numbers. To best understand this new category of real numbers, students need to make comparisons between rational and irrational numbers. By making comparisons, students will develop the understanding of the unique characteristics of each. Typically, the only irrational number students are familiar with is the number pi. However, students have not necessarily connected pi with the characteristics of irrational numbers. Students need to identify the characteristics of irrational numbers as they compare to the rational number system. As students gain more familiarity with irrational numbers, they will be able to estimate the value and create meaning of irrational number solutions as they solve problems involving all real numbers.

All Standard Benchmarks
Classify real numbers as rational or irrational. Know that when a square root of a positive integer is not an integer, then it is irrational. Know that the sum of a rational number and an irrational number is irrational, and the product of a non-zero rational number and an irrational number is irrational.
For example: Classify the following numbers as whole numbers, integers, rational numbers, irrational numbers, recognizing that some numbers belong in more than one category: [math]\frac{6}{3}, \frac{3}{6}, 3.\overline{6}, \frac{\pi }{2}, -\sqrt{4}, \sqrt{10}, -6.7[/math]
Allowable notation: [math]\sqrt{18}[/math]

Compare real numbers; locate real numbers on a number line. Identify the square root of a positive integer as an integer, or if it is not an integer, locate it as a real number between two consecutive positive integers.

 For example: Put the following numbers in order from smallest to largest: [math]2, \sqrt{3}, -4, -6.8, -\sqrt{37}[/math]

Another example: [math]\sqrt{68}[/math] is an irrational number between 8 and 9.

Allowable notation: [math]\sqrt{18}[/math]

Determine rational approximations for solutions to problems involving real numbers.      

For example: A calculator can be used to determine that [math]\sqrt{7}[/math] is approximately 2.65.

Another example: To check that [math]1\frac{5}{8}[/math] is slightly bigger than [math]\sqrt{2}[/math], do the calculation [math]\left ( 1\frac{5}{12} \right )^{2}=\left ( \frac{17}{12} \right )^{2}=\frac{289}{144}=2\frac{1}{144}[/math].

Another example: Knowing that [math]\sqrt{10}[/math] is between 3 and 4, try squaring numbers like 3.5, 3.3, 3.1 to determine that 3.1 is a reasonable rational approximation of [math]\sqrt{10}[/math].

Allowable notation: [math]\sqrt{18}[/math]

Know and apply the properties of positive and negative integer exponents to generate equivalent numerical expressions.

For example:[math]3^{2}\times 3^{(-5)}=3^{(-3)}=\left ( \frac{1}{3} \right )^{3}=\frac{1}{27}[/math]

Allowable notation: -x2, (-x)2, -32, (-3)2 Expressions may be numeric or algebraic.

Express approximations of very large and very small numbers using scientific notation; understand how calculators display numbers in scientific notation. Multiply and divide numbers expressed in scientific notation, express the answer in scientific notation, using the correct number of significant digits when physical measurements are involved.

For example: (4.2 × 104) ×(8.25 ×103) = 3.465 × 108, but if these numbers represent physical measurements, the answer should be expressed as 3.5 × 108 because the first factor,4.2 × 104, only has two significant digits.

Benchmark Cluster 
Benchmark Group B - Integer Exponents and Scientific Notation

Know and apply the properties of positive and negative integer exponents to generate equivalent numerical expressions.

For example: [math]3^{2}\times 3^{(-5)}=3^{(-3)}=\left ( \frac{1}{3} \right )^{3}=\frac{1}{27}[/math] 

Allowable notation: -x2, (-x)2, -32, (-3)2 Expressions may be numeric or algebraic.

Express approximations of very large and very small numbers using scientific notation; understand how calculators display numbers in scientific notation. Multiply and divide numbers expressed in scientific notation, express the answer in scientific notation, using the correct number of significant digits when physical measurements are involved.

For example: (4.2 × 104) ×(8.25 ×103) = 3.465 × 108, but if these numbers represent physical measurements, the answer should be expressed as 3.5 × 108 because the first factor,4.2 × 104, only has two significant digits.

What students should know and be able to do [at a mastery level] related to these benchmarks:
  • Realize that scientific notation is a way to write very large or very small numbers in a readable manner developed by scientists. Students will know that the number is written with a coefficient between 1 and 10 and multiplied by a power of ten;
  • Be able to convert from scientific notation to standard form by interpreting the power of ten being multiplied by the coefficient;
  • Be able to multiply and divide numbers that are written in scientific notation. They will be able to put the answer in proper scientific notation;
  • Recognize scientific notation when it appears on calculator displays;
  • Be able to know and apply the properties of exponents to generate equivalent expressions. This includes multiplying, dividing, raising a power to a power, converting negative exponents, and zero power rule;
  • Be able to interpret scientific notation when it shows up on their calculator. For example, 2.345E6 is 2.345 × 106.
Work from previous grades that supports this new learning includes:
  • Understand how to write numbers in exponential form. For example, 2 × 2 × 2 can be written as 23;
  • Understand scientific notation from exposure in science courses and possibly math courses;
  • Know that any number divided by itself (x/x) simplifies to 1 (note: x cannot be zero).



Student Misconceptions 
Student Misconceptions and Common Errors
  • Students often mistake the exponent as the number of zeros to put on the end of the coefficient instead of realizing it represents the number of times they should multiply by ten.
  • When multiplying or dividing numbers that are given in scientific notation, in which the directions say to write the answer in scientific notation, sometimes students forget to double check that the answer is in correct scientific notation.
  • When performing calculations on a calculator, in which the number transforms to scientific notation, students sometimes overlook the last part of the number showing scientific notation part and just notice the first part of the number, ignoring the number after E.
  • Students will sometimes multiply the base and the exponent. For example, 26 is not equal to 12, it's 64.
  • When writing numbers in scientific notation, students may interpret the negative exponent as a negative number.
  • Students will try and use a rule of exponents when adding or subtracting. For example,

  • Students will try and use the rules of exponents for multiplying or dividing when the bases are not the same. For example,

  • Students often mix up the rules of operating with exponents. Students will multiply exponents when the operation is multiplication. Students will divide when the operation is division. Students will take the exponent to the power when the power is taken to a power.
  • When simplifying expressions involving power to a power, students will sometimes mistakenly "distribute" an exponent over addition or subtraction. For example, for (3 + 4)2, students sometimes will say this equals 32 + 42 = 9 + 16 = 25 when actually order of operations adds 3 + 4 first and results in 72 = 49, not 25.
  • Students sometimes do not realize (or forget) that any number to the zero power is 1. They commonly state that any number to the zero power is 0.
  • Students forget about significant digits if calculating with numbers written in scientific notation.


In the Classroom

This vignette uses a multiplication problem - figuring out the weight of all of the insects in the world - to learn about multiplying exponents.

Teacher: Today I have a question for you. What is the total weight of the world's insect population?

Student: How are we supposed to know?

Teacher: What information would you need to know to figure it out? What questions would you ask?

Student: Well...How many insects are in the world?

Student: What is the weight per insect?

Student: What is the birth rate and death rate?

Student: Yeah, because I kill bugs all the time.

Teacher: In his book, Six-Legged Science, Brian Hocking estimates that there are

1 × 1018 insects in the world.

Student: Why did you write it like that?

Teacher: I wrote in scientific notation because it is a more concise way of writing a very large number. If I wrote one quintillion in standard form, it would look like this: 1,000,000,000,000,000,000.

Student: We still need to know the weight per insect. Then we can just multiply the total insects by the average weight.

Teacher: Any guesses on the average weight?

Student: Well, it has to be really light. Way less than a pound. Maybe even less than a gram. 

Teacher: To simplify things for right now, I am going to tell you that the average weight for an insect is about 2.5 × 10-3grams.

Student: What does the negative 3 as an exponent mean?

Student: An insect is really small. It weighs a lot less than a gram. So doesn't the negative exponent tell us to move the decimal point to the right instead of the left?

Teacher: Some of you may know some rules about moving the decimal place left and right. What do we know about mathematics that allows us to "move" the decimal place?

Student: Well, since the base number is ten, aren't we just repeated multiplying by 10? When we multiply by 10, we move place value, and decimal points help us hold place value.

Teacher: If the positive exponent makes the coefficient get bigger by multiplying by 10, what do you predict the negative exponents do?

Student: We know it has to represent a smaller number and if it moves the decimal to make it smaller, I think the negative would tell us to divide by 10.

Teacher: So let's look at 2.5 × 10-3 grams again. How could I write it in standard form?

Student: Just move the decimal place to the left three times.

Teacher: Why can you do that? What are you really doing?

Student: Oh, she's dividing by 10 three times. So she gets 25 ten thousandths.

Teacher: How could you show that process without just moving the decimal?

Student: Here's one way. 

Student Example 1student example 1

Student: I did it like this.

Student Example 2

Student example 2

Teacher: OK, so now we have what we need. You all told me earlier we could multiply the total insects by their average weight to find the weight of the total insect population. How do you multiply when the number is written in scientific notation?

(1 × 1018)(2.5 × 10-3

Student: It's a big number but can't you just type it in your calculator? I get 2.5E15.

Teacher: What does that mean?

Student: I don't know. 2.5 something....

Teacher: We need to be able to interpret what our calculators tell us, but we also need to be able to work without a calculator.

Student: Hey, I see something. It's all multiplication. So I could write it like this:

1 × 1018 × 2.5 × 10-3. Then I can use my commutative property to change the order and multiply 1 × 2.5 and then 1018 × 10-3. Now I just have to figure out how to multiply the numbers in exponential form. How do I do that?

Teacher: Let's look at this first one. How many times are we multiplying by ten?

Student: 18 times. We could write it out as 10 × 10 eighteen times like this.1/10 18 times

Student: Then 10-3 is like dividing by 10 three time or multiplying by [math]\frac{1}{10}[/math] three times. So I could write it like this: [math]\frac{1}{10(10)(10)}[/math]

Student: So if you put those together we get this?

Teacher: Exactly!

Student: I don't want to write this out like this every time I do scientific notation.

Teacher: We are writing it out so we can see if there is a pattern that might help us find a rule that always works. Let's finish this one and then we will look at some more to see if we can find a way to make this easier.

So, if we simplify this in expanded form, how many times are we left multiplying by 10?

Student: 15. So we would write it as 1015.

Student: So then we would end up with 2.5 × 1015. That's still in scientific notation, too.

Student: Hey, when I got 2.5E15. That must be the calculator's version of scientific notation. The coefficient is the number before the E and the number after the E is the exponent or how many times I multiply by 10. Cool!

Student: So the approximate total weight of all the insects would be 2.5 × 1015 grams.

Student: How heavy is that? How many grams are in a pound? 

Teacher: That will be your homework tonight. Figure out how many pounds the insects weigh and then relate that to something that weighs a similar amount or a group of something that would weigh that amount.

Teacher: OK, so let's go back to those rules. How do we operate with exponents? Let's start with multiplication. 106 × 103. How would you write each number in expanded form?

How could you write this whole thing in a simpler way?

Student: That's easy. 109, because we multiply by 10 a total of nine times.

Teacher: Let's do one more and see if there is a pattern. Let's use a base number of 3 this time. 38 × 34. Write it out in expanded form and then simplify.

Teacher: Let's look at all three examples. Do you notice any patterns?

Student: Hey, it looks like you can just add the exponents if the base numbers are the same. It even works for the negative exponent. That makes sense because it's really all multiplication, and you just need the total number of times you multiply.

Student: So if you multiply in exponential form, we can simplify by keeping the base the same and adding the exponents. That would work when we are multiplying in scientific notation then too. Then I wouldn't have to write out all those numbers.

Student: What if we were dividing instead of multiplying?

Teacher: We kind of already did that when we were working with our insect problem. Because of the exponent of -3, it was very similar to dividing. Look back at that example.


When we wrote it this way, it was the same as [math]\frac{10^{18}}{10^3}[/math]. What happened?

Student: Since ten divided by ten is one, it's kind of like we lose those tens in the denominator. You keep dividing and then whatever you have left over is the simplified answer.

Teacher: Let's try this one: [math]\frac{10^{12}}{10^{8}}[/math]. Write it out in expanded form and then we will look at what is left over.

Student: I did it this way.

Teacher: Let's look at one more that uses a different base than 10. Try this one: [math]\frac{5^{7}}{5^{5}}[/math].

Once you get an answer, look back at the others and come up with a rule that will work any time we divide numbers in exponential form.

Teacher: Here are our three examples. What do you notice?

Student: I notice that the bases have to be the same just like with multiplication.

Student: If the bases are the same and we just look at the exponents, I think we can just subtract the exponents to simplify.

Student: That makes sense, because as we divide out the base numbers it's like we lose one more that we multiply by. We are kind of crossing them out instead of adding them up like we did in multiplication.

Student: OK, just so I know I have this. If we divide in exponential form, if the bases are the same we can subtract the exponents to simplify number?

Teacher: You got it! What happens if you forget these rules?

Student: Well, if you just write it out in expanded form, you'll get the same answer. That will help me remember if I get stuck.

Teacher: In your notebooks, make two sections. One for multiplication and one for division. In each section, make up your own example, show it written out in expanded form and write a sentence explaining the rule.


Instructional Notes 
Teacher Notes
  • When converting from scientific notation to standard notation, avoid teaching students to just move the decimal. Teach students to repeatedly multiply and divide by ten as determined by the exponents.
  • When using scientific notation, use correct vocabulary with the students. In a × 10n, where 1 ≤ a < 10 and n is an integer, a is called the coefficient or mantissa and n is the exponent.
  • When teaching multiplying and dividing in scientific notation, link it to the rules of exponents.

                        Example:  (3.5 × 10-3)(7 × 105)

                        =  (3.5 × 7)(10-3 × 105) by the commutative property of multiplication

                        =  24.5 × 10by product of powers rule of exponents

                        = (2.45 × 101) × 102

                        = 2.45 × 103 by product of powers rule of exponents

  • In order to help students interpret the E on their calculators, ask students to record the number in scientific notation on their papers instead of writing the number with the E.
  • When introducing students to rules of exponents, start by writing each expression in expanded notation, allowing students to see the connection between the rule and the actual math. This will support their understanding of the rule and make it unnecessary for them to memorize the rules.


  • After finding the patterns of multiplying numbers in exponential form, it is helpful for students to write the rules in words so they are explaining what they are doing to simplify the expression.
  • Connect negative exponents to the rules of dividing exponents. Students will then be able to make a connection as to why negative exponents "flip" the base number.

Example: Simplify [math]3^{2}\div 3^{3}[/math]

[math]3^{2}\div 3^{3}=9\div 27=\frac{9}{27}=\frac{1}{3}[/math]
[math]3^{2}\div 3^{3}=3^{2-3}=3^{-1}=\frac{1}{3}[/math]
[math]\frac{3^{2}}{3^{3}}=\frac{3\cdot 3 \cdot 1}{3\cdot 3\cdot 3}=\frac{1}{3}=3^{-1}[/math]

Source: http://www.mathnstuff.com/math/spoken/here/2class/210/laws.htm

  • Connect the power of 0 to the rules of dividing exponents. Students will be able to see that a power to zero is like a number divided by itself, therefore equaling 1.

Example: Simplify [math]3^{2}\div 3^{2}[/math]

[math]3^{2}\div 3^{2}=9\div 9=1[/math]
[math]3^{2}\div 3^{2}=3^{2-2}=3^{0}=1[/math]
[math]\frac{3^{2}}{3^{2}}=\frac{3\cdot 3}{3\cdot 3}=1[/math]

Source: http://www.mathnstuff.com/math/spoken/here/2class/210/laws.htm

  • Make sure the students understand that when a term does not have an exponent it is assumed to be 1.

      Examples: 3 = 31         y = y1

  • Have students realize the difference between a negative coefficient and a negative exponent in scientific notation. A negative sign in the exponent means that the number is between 0 and 1. A negative sign before the number means that it is less than 0.
  • Be specific when working with exponents to include examples that involve negative values. Remind students about order of operations when working with evaluating or simplifying exponents.
    Example -32 or (-3)2 -x2 or (-x)2
  • Students may have worked with significant digits (significant figures) in a science class before. Significant digits are important to look at with measurements because they show the degree of accuracy with which the measurement was obtained. There is a difference between the numbers 4 × 105 which has one significant digit, and 4.0 × 105, which has two significant digits, even though their numeric values are the same. When multiplying or dividing, round the final result to the least number of significant figures that any one term had.

Example: (4.64 × 103)(1.7 × 105) = 7.888 × 10but if these numbers are measurements, we need to be concerned about significant digits. 4.64 × 103 has 3 significant digits and 1.7 × 105  has 2 significant digits so the product can only have 2 significant digits. Therefore the correct answer is 7.9 × 108.


Instructional Resources 

Understanding large numbers
This website helps students visualize large quantities, such as a million, billion and trillion, by showing illustrations of huge stacks of pennies.

Additional Instructional Resources

Powers of Ten video

Big numbers picture book
Schwartz, D.M. (2004) How Much Is A Million? New York, NY: HarperCollins.

Significant digits
This website explains the logic and importance of significant digits, including examples of scientific notation.

Scientific notation videos
This website offers videos about writing numbers in scientific notation, as well as multiplying and dividing scientific notation numbers.

Rules for significant digits

New Vocabulary 

scientific notation: a number in scientific notation is expressed as a × 10n, where 1 ≤ a < 10 and n is an integer.

significant digits (significant figures): for measured numbers, significant digits relate the certainty of the measurement. As the number of significant digits increases, the more certain the measurement.

standard form (standard notation): The general mathematical form to write a number.

Professional Learning Communities 
Reflection - Critical Questions regarding the teaching and learning of these benchmarks
  • How can we ensure that students understand why the decimal point moves to the right/left in converting with scientific notation?
  • What mistakes are students making when finding equivalent expressions using the rules of exponents and how can those mistakes be avoided in the future?
  • In what ways can we make these rules more relevant to students?
  • Multiplying by 10: Scientific notation
    This website provides Minnesota Region 11 training session on multiplying by 10 done by Dr. Terry Wyberg, relating this basic concept of multiplying by 10 to how we teach scientific notation.


1. Simplify.

[display]\frac{1.2\times 10^{-6}}{4.8\times 10^{4}}[/display]

A.  2.5 × 10-2

B.  2.5 × 10-9

C.  2.5 × 10-10

D.  2.5 × 10-11

DOK Level: 2

Answer: D

Source: Minnesota MCA Item Sampler 8th grade

2.  Mr. Loya stated that the circumference of Earth at the equator is 24,902.4 miles.  Which expression represents this number in scientific notation?

A.  2.49024 × 104

B.  24.9024 × 104

C.  249.024 × 104

D.  2.49024 × 105

DOK Level: 1

Answer: A

Source: 2006 Grade 8 TAKS (Texas Assessment of Knowledge and Skills) Sample Test


3.  Which expression is equivalent to x6x2?

A.  x4x3

B.  x5x3

C.  x7x3

D.  x9x3

DOK Level: 1

Answer: B

Source: Algebra I California Standardized Released Test (See downloads in Resources section)




A.  [math]2x^{4}[/math]

B.  [math]\frac{1}{2x^{4}}[/math]

C.  [math]\frac{1}{5x^{4}}[/math]

D.  [math]\frac{x^{4}}{5}[/math]

DOK Level: 2

Answer: B

Source: Algebra I California Standardized Released Test (See downloads in Resources section)


[display]\frac{4^{2}\cdot 3^{5}\cdot 2^{4}}{4^{3}\cdot 3^{5}\cdot 2^{2}}=[/display]

A.  [math]\frac{4}{2}[/math]

B.  [math]\frac{3}{2}[/math]

C.  [math]1[/math]

D.  [math]\frac{1}{2}[/math]

DOK Level: 2

Answer: C

Source: California 7th grade Standardized Released Test (See downloads in Resources section)

6.  The length of a room is 5.048 × 102 cm.  Which number is equivalent to this length?

A.  0.005048 cm

B.  0.05048 cm

C.  504.8 cm

D.  504,800 cm

DOK Level: 1

Answer: C

Source: California 7th grade Standardized Released Test (See downloads in Resources section)

7.  Simplify.

(4x)2 - 4x3

A.  x-1

B.  12x-1

C.  16x2 - 4x3

D.  16x2 - 64x3

DOK Level: 2

Answer: C

Source: Minnesota 8th Grade MCA Item Sampler


Struggling Learners 
  • Multiplying and dividing in scientific notation
    This website provides resources to help students better understand how to multiply and divide scientific notation, such as this graphic organizer. (See Downloads in Resources section)

Multiplying & Dividing #'s in Scientific Notation


English Language Learners 
  • Differentiate for students still developing understanding of the content and for those that demonstrate advanced understanding by the size of numbers used and by the number of problems learners complete.
  • Students that experience success early in the lesson can research situations when scientific notation would be helpful in calculations and when it would not make sense to use it. Once this evaluation has taken place, students can research specific examples, such as space explorations and exponential growth or decay, to back up their evaluation.
  • For students in need of intense intervention, present models of numbers using base-ten blocks or visual representations of base-ten blocks. Begin by having students represent whole numbers with the models or visual representations. Relate the size of the blocks to the exponent; a cube (1 unit) representing the zero power, a rod (10units) representing the first power, a flat (100 units) representing the second power, and a large cube (1000 units) representing the third power. Scaffold understanding by having them represent numbers in scientific notation. Although the blocks have limitations, they provide access to understanding the basis of the concept for a variety of learning preferences.
    Source: Ohio Department of Ed
  • ELL students should be given numerous opportunities to write about the mathematics concepts they are learning. Journal entries, for example, provide opportunities for the students to crystallize their thinking about concepts and for the teacher to check for understanding. Students who have limited English language skills should be allowed to write in their first language initially and should not be penalized for spelling or grammar errors. Have students journal about the rules of exponents and the process for converting scientific notation to standard form and back again.
Extending the Learning 
  • The national debt problem
    This document works on multiplying and dividing numbers written in scientific notation when dealing with a real life situation.
  • Students can examine fractional exponents and their relationship to radicals.


Classroom Observation 

Administrative/Peer Classroom Observation

Students are: (descriptive list)

Teachers are: (descriptive list)

writing the numbers out in expanded form in order to learn the rules when operating with exponents.

reminding students to repeatedly multiply and divide by ten to convert from scientific notation to standard form.

writing sentences to describe the rules of exponents.

connecting the multiplying and dividing of scientific notation to rules of exponents.

operating with scientific notation in context of a real world problem.

providing students with real world contexts when they are multiplying and dividing in scientific notation.

writing expressions in expanded form to understand the reasoning behind the rules of exponents.

helping students understand the "why" of exponent rules by having students write out numbers in expanded form instead of memorizing rules.

working with positive and negative exponents.