Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3 × 3 = 3 = 1/3 = 1/27.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Exponents are shorthand for repeated multiplication, and a handful of properties fall straight out of that meaning. Writing 2³ × 2⁴ in full gives seven factors of 2, so the exponents add. Dividing 5⁶ by 5² cancels two factors, so the exponents subtract. Raising a power to a power, (3²)³, multiplies the exponents because you have three groups of two factors. Grade 8 students should be able to derive each rule from expanded form, not just recite it.
The new territory is zero and negative exponents. Following the pattern 2³ = 8, 2² = 4, 2¹ = 2, each step divides by 2, so 2⁰ = 1, 2⁻¹ = 1/2 and 2⁻³ = 1/8. A negative exponent therefore means a reciprocal, not a negative number. Students use all of these properties to rewrite numerical expressions in equivalent forms, for instance showing that 3² × 3⁻⁵ = 3⁻³ = 1/27, or that 10⁴ ÷ 10⁶ is the same as 1/100.
Students write 2³ × 2⁴ = 4⁷. Expanding to (2 × 2 × 2) × (2 × 2 × 2 × 2) shows the base stays 2 and only the count of factors changes.
5⁻² is not -25. It means 1/5², which is 1/25, a small positive number. The pattern of repeated division makes this clear.
Because the exponent is zero, many students expect the value to be zero. Continuing the halving pattern down from 2³ lands on 2⁰ = 1.
(3²)⁴ is four copies of 3², which is eight factors of 3, so the exponents multiply to give 3⁸, not 3⁶.
Write 3² × 3⁻⁵ as a single power of 3 and then as a fraction.
Answer: 3² × 3⁻⁵ = 3⁻³ = 1/27
Have students build a table of powers of 2 or 10 going downward from 2⁵ and say out loud what happens at each step. Once they see that each row halves the one above, 2⁰ = 1 and 2⁻¹ = 1/2 feel inevitable rather than arbitrary. Keep expanded form available as the fallback whenever a student is unsure which rule applies.
Questions on this standard are often select-all items (which expressions are equivalent to 4⁻³?) or short simplifications. Insist on numerical bases at this grade, since 8.EE.A.1 is about numerical expressions, and let algebraic bases wait for high school.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: C) 2⁷
Same base, so add the exponents: 3 + 4 = 7, giving 2⁷. The base stays 2.
Answer: 1 (also accepted: one)
Any nonzero number raised to the power 0 equals 1.
Answer: 1/16
4⁻² = 1/4² = 1/16.
Answer: D) 5⁶
A power of a power multiplies the exponents: 2 × 3 = 6, so (5²)³ = 5⁶.
Answer: 100
Subtract the exponents: 10⁵⁻³ = 10² = 100.
Answer: B) 1/8
2⁻³ = 1/2³ = 1/8. A negative exponent gives a reciprocal, not a negative number.
Answer: 9
Add the exponents: 4 + (-2) = 2. 3² = 9.
A full lesson with slides, activities and an exit ticket on properties of integer exponents, pitched to grade 8 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 8.EE.A.1 with an answer key, ready in about a minute.
Make a worksheet →Turn properties of integer exponents into a quiz students answer online that marks itself, with a class summary for you.
Build a test →Using the quotient rule, 5³ ÷ 5³ = 5⁰, and any nonzero number divided by itself is 1. So 5⁰ must be 1.
The standard itself is about numerical expressions. Students often see variables in practice, but the expectation in grade 8 is fluent work with numerical bases.