Grade 8 · Real Numbers & Exponents · MAT-08-NS-006
Apply Integer Exponent Rules
Represent and solve apply integer exponent rules problems using precise mathematical notation, multiple representations, and justified reasoning.
Learning objectives
What you should be able to do
- Represent and solve apply integer exponent rules problems using precise mathematical notation, multiple representations, and justified reasoning.
- Interpret and verify apply integer exponent rules results in context, including units, constraints, and reasonableness.
Prerequisite check
Make sure the foundation is ready.
Grade 8 uses this prerequisite as active knowledge in a more formal representation.
Grade 8 uses this prerequisite as active knowledge in a more formal representation.
Learn
Build the idea from meaning, not memorization.
Meaning and structure
Apply Integer Exponent Rules belongs to real numbers & exponents. Identify the governing property or relationship before choosing a procedure.
Multiple representations
Represent apply integer exponent rules numerically and symbolically, then connect it to the Grade 8 Math Box. Tables, graphs, equations, coordinate models, diagrams, and verbal descriptions must agree.
Verification and interpretation
Check with substitution, an inverse operation, an equivalent representation, geometric invariants, estimation, or context. State whether the result is exact or approximate and preserve relevant units and constraints.
Math Box · Interactive lesson
Touch the math. Change it. See what stays true.
Use the interactive model before and after the worked examples. Change the inputs, make a prediction, then use the model to test whether your reasoning holds.
Exponent Rules Lab
Change the base and integer exponent. Negative exponents expose reciprocal structure.
- Model apply integer exponent rules in the Math Box.
- Change one input and predict which quantities or invariants should change.
- Explain what the visual, equation, and numerical result say about the same relationship.
Worked examples
Twenty different ways to see the concept work.
Simplify 2^1 × 2^2.
2^3When multiplying powers with the same base, add exponents.
Simplify 3^2 × 3^4.
3^6When multiplying powers with the same base, add exponents.
Simplify 4^3 × 4^6.
4^9When multiplying powers with the same base, add exponents.
Simplify 5^4 × 5^3.
5^7When multiplying powers with the same base, add exponents.
Simplify 6^5 × 6^5.
6^10When multiplying powers with the same base, add exponents.
Simplify 7^1 × 7^2.
7^3When multiplying powers with the same base, add exponents.
Simplify 8^2 × 8^4.
8^6When multiplying powers with the same base, add exponents.
Simplify 9^3 × 9^6.
9^9When multiplying powers with the same base, add exponents.
Simplify 2^4 × 2^3.
2^7When multiplying powers with the same base, add exponents.
Simplify 3^5 × 3^5.
3^10When multiplying powers with the same base, add exponents.
Simplify 4^1 × 4^2.
4^3When multiplying powers with the same base, add exponents.
Simplify 5^2 × 5^4.
5^6When multiplying powers with the same base, add exponents.
Simplify 6^3 × 6^6.
6^9When multiplying powers with the same base, add exponents.
Simplify 7^4 × 7^3.
7^7When multiplying powers with the same base, add exponents.
Simplify 8^5 × 8^5.
8^10When multiplying powers with the same base, add exponents.
Simplify 9^1 × 9^2.
9^3When multiplying powers with the same base, add exponents.
Simplify 2^2 × 2^4.
2^6When multiplying powers with the same base, add exponents.
Simplify 3^3 × 3^6.
3^9When multiplying powers with the same base, add exponents.
Simplify 4^4 × 4^3.
4^7When multiplying powers with the same base, add exponents.
Simplify 5^5 × 5^5.
5^10When multiplying powers with the same base, add exponents.
Practice
Guided practice with hints
01Simplify 6^1 × 6^2.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: 6^3
02Simplify 7^2 × 7^4.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: 7^6
03Simplify 8^3 × 8^6.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: 8^9
04Simplify 9^4 × 9^3.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: 9^7
05Simplify 2^5 × 2^5.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: 2^10
06Simplify 3^1 × 3^2.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: 3^3
Practice
Independent practice
01Simplify 4^2 × 4^4.
Answer: 4^6
02Simplify 5^3 × 5^6.
Answer: 5^9
03Simplify 6^4 × 6^3.
Answer: 6^7
04Simplify 7^5 × 7^5.
Answer: 7^10
05Simplify 8^1 × 8^2.
Answer: 8^3
06Simplify 9^2 × 9^4.
Answer: 9^6
07Simplify 2^3 × 2^6.
Answer: 2^9
08Simplify 3^4 × 3^3.
Answer: 3^7
Common mistakes
Learn to catch the error, not just the answer.
Repeating nonterminating decimals are rational.
Product and quotient exponent rules require the same base and multiplication or division.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Represent extremely large and small measurements with scientific notation.
- Approximate roots and irrational quantities when decimal estimates are needed.
Challenge problems
Simplify 4^5 × 4^5.
4^10
Simplify 5^1 × 5^2.
5^3
Simplify 6^2 × 6^4.
6^6
Simplify 7^3 × 7^6.
7^9
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 8 Algebra & Modeling Sprint
Solve apply integer exponent rules problems and justify each result with a valid verification method.
Practice
Mastery check
01Simplify 8^4 × 8^3.
Answer: 8^7
02Simplify 9^5 × 9^5.
Answer: 9^10
03Simplify 2^1 × 2^2.
Answer: 2^3
04Simplify 3^2 × 3^4.
Answer: 3^6
05Simplify 4^3 × 4^6.
Answer: 4^9
06Simplify 5^4 × 5^3.
Answer: 5^7
07Simplify 6^5 × 6^5.
Answer: 6^10
08Simplify 7^1 × 7^2.
Answer: 7^3
Terminology
Words to know
- rational number
- A real number expressible as a ratio of integers with nonzero denominator.
- irrational number
- A real number that cannot be written as a ratio of integers.
- exponent
- A number indicating repeated multiplication by a base.
- scientific notation
- A representation a × 10^n where 1 ≤ |a| < 10 and n is an integer.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
California Mathematics Framework (2023)
Grade 8 instructional guidance, mathematical practices, modeling, and progression toward high-school mathematics.Common Core State Standards for Mathematics — Grade 8
Grade-level expectations for real numbers, equations, functions, geometry, and statistics.Principles to Actions: Ensuring Mathematical Success for All
Research-informed emphasis on reasoning, representations, discourse, productive struggle, and conceptual understanding.