8.EE.A.1Common CoreMathExpressions and EquationsGrade 8
8.EE.A.1: Properties of Integer Exponents
In plain English: 8.EE.A.1 is the Common Core grade 8 math standard that asks students to know and use the properties of integer exponents, including zero and negative exponents, to rewrite numerical expressions in equivalent forms. For example, 3² × 3⁻⁵ = 3⁻³ = 1/27. It is part of Expressions and Equations in Grade 8 Math and leads to scientific notation.
Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3² × 3⁻⁵ = 3⁻³ = 1/3³ = 1/27.
Common Core State Standards for Mathematics · Domain: Expressions and Equations (EE) · Cluster: Work with radicals and integer exponents. Also written as 8.EE.1 · Official standard
In grade 6, students wrote powers such as 2⁵ with whole-number exponents. In this lesson they extend exponents to all integers (the whole numbers, their opposites and zero), so exponents can also be 0 or negative. Students learn the properties of integer exponents: rules that are true for every nonzero base and every pair of integer exponents. They use the rules to write equivalent numerical expressions, which are expressions made of numbers that have the same value, such as 3² × 3⁻⁵ and 1/27.
The lesson starts from a pattern: each time the exponent goes down by 1, the value is divided by the base. That pattern explains why 2⁰ = 1 and why a negative exponent gives a reciprocal (1 divided by the number), not a negative number. Students then practice the product, quotient and power rules, work through the official example 3² × 3⁻⁵ = 3⁻³ = 1/3³ = 1/27, and use powers of 10 to convert metric units.
Learning Objectives
By the end of this lesson, students will be able to:
State the product, quotient, power of a power and power of a product properties, and the meaning of zero and negative exponents
Explain with a pattern why a nonzero number to the zero power is 1 and why a negative exponent gives a reciprocal
Rewrite a numerical expression with integer exponents as a single power (one base with one exponent) and find its value as a whole number or fraction
Find and correct errors such as multiplying the bases or treating a negative exponent as a negative number
Prior Knowledge Required
Students should already be comfortable with:
Writing and evaluating expressions with whole-number exponents, such as 4³ = 64 6.EE.A.1
Explaining patterns in the number of zeros when multiplying by powers of 10 5.NBT.A.2
Adding and subtracting integers, including negative numbers 7.NS.A.1
Multiplying and dividing fractions and other rational numbers 7.NS.A.2
Recall the words from grade 6. In 2⁴, the base is 2 (the number being multiplied), the exponent is 4 (how many times the base is used as a factor), and the whole expression is a power. Copy this table on the board and give pairs three minutes.
Powers of 2: fill in the last three rows by following the pattern.
Power
Value
2⁴
16
2³
8
2²
4
2¹
2
2⁰
?
2⁻¹
?
2⁻²
?
Warm-Up Prompt
"Going down the table, what do you do to the value each time the exponent goes down by 1? Keep the pattern going. What should 2⁰, 2⁻¹ and 2⁻² be?"
Collect answers. Each step down divides by 2, so 2⁰ = 2 ÷ 2 = 1, 2⁻¹ = 1 ÷ 2 = 1/2 and 2⁻² = 1/2 ÷ 2 = 1/4. Some students will guess 2⁰ = 0 or 2⁻¹ = -2. Ask them to check their guess against the pattern: dividing 2 by 2 cannot give 0. Tell students that today they will learn the rules, called properties, that make this pattern work for every base.
Direct Instruction20 minutes
Use Diagram 1 to extend the warm-up pattern in both directions. Then write each property on an anchor chart. A property is a rule that is always true. In each rule, the base is any nonzero number and the exponents m and n are any integers. Show one numerical example for each rule by writing out the factors.
Product of powers (a product is the result of multiplying): with the same base, add the exponents. 4² × 4³ = (4 × 4) × (4 × 4 × 4) = 4⁵. In general, am × an = am + n.
Quotient of powers (a quotient is the result of dividing): with the same base, subtract the exponents. 6⁵ ÷ 6² = 6³, because two 6s in the numerator cancel with the two 6s in the denominator. In general, am ÷ an = am - n.
Power of a power: multiply the exponents. (5²)³ = 5² × 5² × 5² = 5⁶. In general, (am)n = amn.
Power of a product: raise each factor to the power. (2 × 3)⁴ = 2⁴ × 3⁴. The same works for a quotient: (2/3)⁴ = 2⁴/3⁴.
Zero exponent: 9³ ÷ 9³ = 1, and the quotient rule gives 9³ ÷ 9³ = 9⁰. So 9⁰ = 1. Any nonzero base to the zero power is 1.
Negative exponent: a negative exponent means the reciprocal of the positive power. 10⁻³ = 1/10³ = 1/1,000. For a fraction, flip it: (1/5)⁻² = 5² = 25. A negative exponent never makes the value negative.
Work through the five examples below. For each one, ask students to name the property before they use it. The official example comes first; Diagram 2 shows it by cancelling factors.
Product of powers (official example)
Write 3² × 3⁻⁵ as a single power, then as a fraction.
Equation: Same base, so add the exponents: 2 + (-5) = -3. 3² × 3⁻⁵ = 3⁻³ = 1/3³ = 1/27
Quotient of powers
Write 7³ ÷ 7⁵ as a single power and find its value.
Equation: The exponents match, so multiply the bases: 5³ × 2³ = (5 × 2)³ = 10³ = 1,000
Metric units with powers of 10
A kilometer is 10³ meters and a millimeter is 10⁻³ meter. How many millimeters are in 1 kilometer?
Equation: Divide: 10³ ÷ 10⁻³ = 10⁶, because 3 - (-3) = 6. There are 10⁶ = 1,000,000 millimeters in a kilometer
Stress two points. First, the properties need the same base (product and quotient) or the same exponent (power of a product); 2³ × 5⁴ cannot be combined into one power. Second, subtracting a negative exponent adds: in the metric example, 3 - (-3) = 6. Students who write 10⁰ there have dropped the sign.
Guided Practice15 minutes
Pairs copy and complete the table, one row at a time. After each row, one pair names the property they used and explains it at the board.
Rewrite each expression as a single power, then find its value.
Expression
Property
Single power
Value
8⁻¹ × 8³
?
?
?
10⁴ ÷ 10⁶
?
?
?
(3²)⁻¹
?
?
?
4⁰ × 4⁻¹
?
?
?
Answers: 8⁻¹ × 8³ = 8² = 64 (product); 10⁴ ÷ 10⁶ = 10⁻² = 1/100 (quotient); (3²)⁻¹ = 3⁻² = 1/9 (power of a power); 4⁰ × 4⁻¹ = 4⁻¹ = 1/4 (product, with 4⁰ = 1). Then show an error to fix together: Sam writes 5² × 5⁴ = 25⁶. He multiplied the bases. The base stays 5, so the answer is 5⁶ = 15,625. Listen for students who multiply exponents in a product, and ask them to write out the factors to check.
Independent Practice15 minutes
Students work alone on five problems, then compare with a partner. (1) 6⁻⁴ × 6⁶. (6² = 36.) (2) 2⁹ ÷ 2¹². (2⁻³ = 1/8.) (3) (5⁻¹)². (5⁻² = 1/25.) (4) 2⁴ × 5⁴. ((2 × 5)⁴ = 10⁴ = 10,000.) (5) Write 1/243 as a power of 3 with a negative exponent. (3⁵ = 243, so 1/243 = 3⁻⁵.)
Closure5-10 minutes
Exit ticket: (1) Write 5⁻¹ × 5⁻¹ as a single power and as a fraction. (5⁻² = 1/25.) (2) In one or two sentences, explain why 12⁰ = 1. (12² ÷ 12² = 1, and the quotient rule gives 12² ÷ 12² = 12⁰.) (3) A classmate says 3⁻² = -9. Explain the mistake. (A negative exponent means a reciprocal: 3⁻² = 1/3² = 1/9.)
Differentiation Strategies
For Struggling Students
Let students write out the factors, such as 4² × 4³ = (4 × 4) × (4 × 4 × 4), before using a rule, until they trust the rule
Keep Diagram 1 on the desk so students can read 2⁰ and the negative powers of 2 from the pattern
Give a property card with one numerical example per rule, and have students circle the base and the exponents in each problem first
For Advanced Students
Ask students to explain why the product rule forces 5⁻¹ = 1/5 (hint: 5¹ × 5⁻¹ = 5⁰ = 1)
Ask: which is larger, 2⁻¹⁰ or 10⁻², and how can you tell without a calculator? (2¹⁰ = 1,024 is larger than 10² = 100, so 2⁻¹⁰ is smaller)
Preview (beyond this standard): in high school, students give meaning to exponents such as 1/2 in HSN.RN.A.1
Assessment Guidance
What to Look For
Ask students to name the property for each step. Watch for four errors: multiplying the bases (5² × 5⁴ = 25⁶), multiplying exponents in a product (4² × 4³ = 4⁶), treating a negative exponent as a negative value (3⁻² = -9) and treating the zero power as 0. A strong answer rewrites the expression as a single power first and only then finds the value as a whole number or fraction.
02
Classroom Activities
3 Activities
1
Equivalent Expression Match
15 minPairs
Each pair gets 12 cards. Every card shows an expression, a single power or a value. Pairs sort the cards into 4 groups of 3 equivalent cards: one expression, its single power and its value.
The 12 Cards (with answers)
Group A: 2⁻² × 2⁻³, 2⁻⁵ and 1/32
Group B: 10⁸ ÷ 10⁵, 10³ and 1,000
Group C: (3⁻¹)⁻³, 3³ and 27
Group D: 7⁴ × 7⁻⁴, 7⁰ and 1
Shuffle the cards before handing them out. The answers are for the teacher.
Procedure
Lay out all 12 cards face up
Start with the expression cards and write the property used on a sticky note next to each group
Check each value by writing out the factors for at least one group
Discussion Questions
Only one group has a value less than 1. Which one, and how can you tell from its single power?
Group C has two negative exponents but a whole-number value. Why?
Group D uses two different exponents but ends at 7⁰. What does that tell you about 7⁴ and 7⁻⁴?
Modification for Distance Learning
Put the 12 cards on a shared slide. Pairs drag each card into one of four boxes and type the property they used.
2
Error Detectives
15 minPairs
Pairs get 6 statement cards. Some statements are right and some are wrong. For each card, pairs decide which, name the property, and rewrite every wrong statement correctly with its value.
The 6 Statement Cards (with answers)
Card 1: 4³ × 4² = 16⁵ (wrong: the base stays 4, so 4⁵ = 1,024)
Each partner judges three cards, then they swap and check each other
For a wrong card, write one sentence that names the mistake
Pairs share one corrected card with the class
Discussion Questions
Four of the six cards are wrong. Which mistake do you think is easiest to make, and why?
After you correct it, Card 2 has the same value as Card 3. Why? (Hint: 8 = 2³.)
Write your own wrong statement for another pair to fix.
3
Powers of Ten Metric Ladder
20 minGroups of 3
Groups build a ladder on a paper strip, one rung for each metric prefix (the start of a unit name, such as kilo- or milli-), labeled with its power of 10 in meters. They use the quotient property to convert units and check three answers with a meter stick.
The Ladder
kilometer: 10³ m
hectometer: 10² m
dekameter: 10¹ m
meter: 10⁰ m
decimeter: 10⁻¹ m
centimeter: 10⁻² m
millimeter: 10⁻³ m
Conversion Questions (with answers)
(a) How many centimeters are in a meter? 10⁰ ÷ 10⁻² = 10² = 100
(b) How many millimeters are in a centimeter? 10⁻² ÷ 10⁻³ = 10¹ = 10
(c) How many millimeters are in a meter? 10⁰ ÷ 10⁻³ = 10³ = 1,000
(d) What part of a kilometer is a centimeter? 10⁻² ÷ 10³ = 10⁻⁵ = 1/100,000
(e) How many decimeters are in a hectometer? 10² ÷ 10⁻¹ = 10³ = 1,000
Procedure
Draw the 7 rungs evenly spaced on the strip, largest unit at the top
For each question, write the division of powers first, then use the quotient property
Check (a), (b) and (c) by counting marks on a meter stick
Challenge Variation
Count the rungs between two units. Why does the number of rungs always equal the exponent of the answer? Groups write the rule in their own words.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: The Powers of 2 Pattern
Moving right, each value is the one before divided by 2, and the exponent goes down by 1. The pattern gives 2⁰ = 1 and shows that a negative exponent gives a fraction, not a negative number.
Diagram 2: The Official Example by Cancelling Factors
3⁻⁵ means 1/3⁵, so 3² × 3⁻⁵ is a fraction with two 3s on top and five 3s on the bottom. After two pairs cancel, 1/3³ = 1/27 is left, which matches adding the exponents.
04
Homework Assignment
~30 min
8.EE.A.1 Homework: Using the Properties of Integer Exponents
Directions: For each expression, first write it as a single power, then give its value as a whole number or a fraction. Name the property you use in each step. Do not use a calculator until you check your answers.
Power of a power and power of a product. (a) (11²)⁻¹ (b) 5³ × 3³ (write it as one power first) (c) (10⁻²)²
Part 2: Combining Properties (Problems 4-6)
Simplify each expression step by step. (a) (5³ × 5⁻⁶) ÷ 5⁻² (b) (6²)³ × 6⁻⁵
Write three different expressions that are each equivalent to 5⁻⁴. Use a different property in each one. Then give the value of 5⁻⁴ as a fraction.
Kai simplified 4⁻² × 4⁵ ÷ 4⁰ like this: "4⁻² × 4⁵ = 16³, and 16³ ÷ 4⁰ = 16³ ÷ 0, which has no answer." Find his two mistakes and give the correct single power and value.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Single Powers
Every expression is rewritten as the correct single power
One or two exponents are wrong
Most exponents are wrong
Values
Every value is correct, and negative exponents give fractions
One or two values are wrong
Most values are wrong or negative exponents give negative values
Properties Named
Names the correct property in each step
Names some properties
No properties named
Error Analysis (Problem 6)
Finds both mistakes and gives the correct single power and value
Finds one mistake
Finds no mistake
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order without a calculator. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
Which statement uses a property of integer exponents correctly?
Answer: A
To multiply powers with the same base, add the exponents: 2² × 2³ = (2 × 2) × (2 × 2 × 2) = 2⁵ = 32. Choice B multiplies the exponents, but 2⁶ = 64, not 4 × 8 = 32. Choice C adds the exponents in a power of a power, but (2²)³ = 2² × 2² × 2² = 2⁶. Choice D divides the exponents; the quotient rule subtracts them, so 2⁶ ÷ 2² = 2⁴.
Question 2 of 20 · Multiple Choice
Write 7⁴ × 7⁻⁹ as a single power.
Answer: C
Same base, so add the exponents: 4 + (-9) = -5, giving 7⁻⁵. Choice A multiplies the exponents, 4 × (-9) = -36. Choice B also multiplies the bases, 7 × 7 = 49. Choice D subtracts, 4 - (-9) = 13.
Question 3 of 20 · Multiple Choice
Write 11⁸ ÷ 11³ as a single power.
Answer: B
Same base, so subtract the exponents: 8 - 3 = 5, giving 11⁵. Choice A adds the exponents. Choice C also divides the bases, 11 ÷ 11 = 1, but the base stays 11. Choice D multiplies the exponents.
Question 4 of 20 · Multiple Choice
Write (6⁻²)³ as a single power.
Answer: D
For a power of a power, multiply the exponents: (-2) × 3 = -6, giving 6⁻⁶. Choice A adds them, -2 + 3 = 1. Choice B cubes the exponent, (-2)³ = -8. Choice C drops the negative sign.
Question 5 of 20 · Multiple Choice
What is the value of 6⁻³?
Answer: A
A negative exponent means the reciprocal: 6⁻³ = 1/6³ = 1/216. Choice B treats the negative exponent as a negative value, -(6³). Choice C multiplies 6 × (-3). Choice D takes the reciprocal of 6 × 3 instead of 6³.
Question 6 of 20 · Multiple Choice
What is the value of 5⁰ + 5⁻¹?
Answer: C
5⁰ = 1 and 5⁻¹ = 1/5, so the sum is 1 + 1/5 = 6/5. Choice A treats 5⁰ as 5: 5 + 1/5 = 26/5. Choice B treats 5⁰ as 0. Choice D treats 5⁻¹ as -5: 1 + (-5) = -4.
Question 7 of 20 · Multiple Choice
Which expression is equal to 1/2⁸?
Answer: B
The reciprocal of a power has the opposite exponent: 1/2⁸ = 2⁻⁸. Choice A is a negative number, -256. Choice C swaps the base and the exponent: 8⁻² = 1/64. Choice D flips the fraction twice: (1/2)⁻⁸ = 2⁸ = 256.
Question 8 of 20 · Multiple Choice
What is the value of 5⁻² × 5⁵?
Answer: D
Add the exponents: -2 + 5 = 3, so the value is 5³ = 125. Choice A multiplies the exponents, 5⁻¹⁰ = 1/9,765,625. Choice B multiplies the base by the new exponent, 5 × 3. Choice C subtracts, 5 - (-2) = 7, and 5⁷ = 78,125.
Question 9 of 20 · Multiple Choice
Which expression is NOT equal to 3⁸?
Answer: A
3² × 3⁴ = 3⁶, because 2 + 4 = 6. The others all equal 3⁸: 10 + (-2) = 8, 4 × 2 = 8 and 12 - 4 = 8. A student who multiplies the exponents in a product gets 2 × 4 = 8 and wrongly thinks choice A is equal to 3⁸.
Question 10 of 20 · Multiple Choice
Which expression is equal to 4² × 5²?
Answer: C
The exponents match, so multiply the bases: 4² × 5² = (4 × 5)² = 20² = 400. Choice A also adds the exponents, which gives 20⁴ = 160,000. Choice B adds the bases instead of multiplying them. Choice D adds both the bases and the exponents.
Question 11 of 20 · Multiple Choice
A small ant has a mass of about 10⁻³ gram. About how many of these ants together have a mass of 10² grams?
Answer: B
Divide the total mass by the mass of one ant: 10² ÷ 10⁻³ = 10⁵, because 2 - (-3) = 5. That is about 100,000 ants. Choice A multiplies instead of dividing (2 + (-3) = -1). Choice C multiplies the exponents. Choice D divides in the wrong order, 10⁻³ ÷ 10², which gives the part of the total that one ant makes up, not a number of ants.
Question 12 of 20 · Multiple Choice
Mia says 12⁻² = -144. Which statement explains her error and gives the correct value?
Answer: D
12⁻² is the reciprocal of 12², which is 1/144, a small positive number. Choice A multiplies 12 × (-2). Choice B moves the negative sign onto the base; (-12)² = 144 is a different expression. Choice C repeats Mia's error.
Question 13 of 20 · Multiple Choice
Which expression has the greatest value?
Answer: C
(1/4)⁻¹ is the reciprocal of 1/4, which is 4. The other values are 4⁻¹ = 1/4, 4⁰ = 1 and 4⁻³ = 1/64. A student who thinks every negative exponent makes a number small picks choice B, 4⁰ = 1.
Question 14 of 20 · Multiple Choice
What is the value of (2/3)⁻²?
Answer: A
Flip the fraction and use the positive exponent: (2/3)⁻² = (3/2)² = 9/4. Choice B ignores the negative sign and gives (2/3)². Choice C makes (2/3)² negative instead of flipping it. Choice D flips it and also makes it negative.
Question 15 of 20 · Short Answer
Write 2⁵ × 2⁻⁹ as a single power. Then write its value as a fraction.
Simplify (7²)⁻¹ × 7³. Show each step and name the property you use.
Power of a power: (7²)⁻¹ = 7⁻². Product of powers: 7⁻² × 7³ = 7¹. The value is 7.
Question 17 of 20 · Short Answer
Ravi says 8⁰ = 0 because "there are no 8s." Use the quotient property to show why 8⁰ = 1.
Any nonzero number divided by itself is 1, so 8² ÷ 8² = 64 ÷ 64 = 1. The quotient property gives 8² ÷ 8² = 82 - 2 = 8⁰. Both are the same expression, so 8⁰ = 1.
Question 18 of 20 · Short Answer
Write 1/343 as a power of 7 with a negative exponent.
7 × 7 × 7 = 343, so 1/343 = 1/7³ = 7⁻³.
Question 19 of 20 · Short Answer
Find the missing integer exponent: 4⁷ × 4? = 4⁻².
The exponents add: 7 + ? = -2, so the missing exponent is -9. Check: 4⁷ × 4⁻⁹ = 47 + (-9) = 4⁻².
Question 20 of 20 · Short Answer
Is 9⁻¹ × 9⁻¹ greater than, less than or equal to 9⁻¹? Explain without a calculator.
Less than. 9⁻¹ × 9⁻¹ = 9⁻² = 1/81, and 9⁻¹ = 1/9. Multiplying 1/9 by another 1/9 makes it smaller, and 1/81 is less than 1/9.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 8.EE.A.1 mean?
8.EE.A.1 means students know the properties of integer exponents and use them to rewrite numerical expressions in equivalent forms. The exponents can be positive, negative or zero. The official example is 3² × 3⁻⁵ = 3⁻³ = 1/3³ = 1/27: add the exponents, then read the negative exponent as a reciprocal.
What are the properties of integer exponents in 8.EE.A.1?
There are five rules and two meanings. With the same base, multiply by adding exponents and divide by subtracting them. A power of a power multiplies the exponents. A power of a product or quotient applies the exponent to each part. Any nonzero base to the zero power is 1, and a negative exponent means the reciprocal of the positive power.
Why doesn't a negative exponent make a number negative?
Because the exponent counts divisions, not a sign. In the pattern 2³ = 8, 2² = 4, 2¹ = 2, 2⁰ = 1, each step divides by 2, so the next values are 1/2, 1/4 and 1/8. Those are 2⁻¹, 2⁻² and 2⁻³. They get smaller but stay positive.
Why is any number to the zero power equal to 1?
Because dividing a power by itself gives 1, and the quotient rule turns that division into a zero exponent. For example, 5⁴ ÷ 5⁴ = 1 and 5⁴ ÷ 5⁴ = 5⁰. The rule is for nonzero bases only; 0⁰ is not defined in grade 8 work.
Can I use the exponent rules when the bases are different?
Only when the exponents are the same. 2³ × 5³ can be written as (2 × 5)³ = 10³, but 2³ × 5⁴ cannot be written as one power. In that case, find each power and multiply the values.
Is 8.EE.A.1 the same as scientific notation?
No, but it comes right before it. 8.EE.A.1 teaches the exponent rules. The next standards, 8.EE.A.3 and 8.EE.A.4, use those rules with powers of 10 to write and compute with very large and very small numbers in scientific notation.
What mistakes do students make with exponent properties?
A few errors come up again and again. Students multiply the bases (writing 16⁵ for 4³ × 4²), multiply the exponents in a product, treat a negative exponent as a negative number, or say a number to the zero power is 0. Writing out the factors for one small case catches each of these.
Do students need to use variables for 8.EE.A.1?
No. The standard is about numerical expressions, so every problem uses numbers. Teachers often write the rules with letters, such as am × an = am + n, to state them in general. Simplifying expressions with variables comes later, in Algebra I.
How is 8.EE.A.1 usually tested?
Tests usually ask students to pick or write an expression equivalent to a given one, to find the value of an expression with zero or negative exponents, or to find the mistake in someone's work. Answers are often fractions, so students should be ready to write 1/49 rather than a decimal.
How can parents help with exponent rules at home?
Ask your child to explain a rule by writing out the factors, for example why 2³ × 2⁴ has seven 2s. Metric units are a good everyday use: a millimeter is 10⁻³ meter, and a kilometer is 10³ meters. Ask how many millimeters are in a meter and how the exponents show it.
07
Related Standards
5 standards
These standards connect to 8.EE.A.1: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
6.EE.A.1Prerequisite
Write and evaluate numerical expressions with whole-number exponents
