Grade 8 · Real Numbers & Exponents · MAT-08-NS-008
Use Powers of Ten and Scientific Notation
Represent and solve use powers of ten and scientific notation problems using precise mathematical notation, multiple representations, and justified reasoning.
Learning objectives
What you should be able to do
- Represent and solve use powers of ten and scientific notation problems using precise mathematical notation, multiple representations, and justified reasoning.
- Interpret and verify use powers of ten and scientific notation 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.
Learn
Build the idea from meaning, not memorization.
Meaning and structure
Use Powers of Ten and Scientific Notation belongs to real numbers & exponents. Identify the governing property or relationship before choosing a procedure.
Multiple representations
Represent use powers of ten and scientific notation 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.
Scientific Notation Lab
Keep the coefficient between 1 and 10 while changing the power of ten.
- Model use powers of ten and scientific notation 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.
Write 1.0 × 10^2 in standard notation.
100A positive power of ten moves the decimal point right by the exponent count.
Write 1.1 × 10^3 in standard notation.
1100A positive power of ten moves the decimal point right by the exponent count.
Write 1.2 × 10^4 in standard notation.
12000A positive power of ten moves the decimal point right by the exponent count.
Write 1.3 × 10^5 in standard notation.
130000A positive power of ten moves the decimal point right by the exponent count.
Write 1.4 × 10^6 in standard notation.
1400000A positive power of ten moves the decimal point right by the exponent count.
Write 1.5 × 10^7 in standard notation.
15000000A positive power of ten moves the decimal point right by the exponent count.
Write 1.6 × 10^8 in standard notation.
160000000A positive power of ten moves the decimal point right by the exponent count.
Write 1.7 × 10^2 in standard notation.
170A positive power of ten moves the decimal point right by the exponent count.
Write 1.0 × 10^3 in standard notation.
1000A positive power of ten moves the decimal point right by the exponent count.
Write 1.1 × 10^4 in standard notation.
11000A positive power of ten moves the decimal point right by the exponent count.
Write 1.2 × 10^5 in standard notation.
120000A positive power of ten moves the decimal point right by the exponent count.
Write 1.3 × 10^6 in standard notation.
1300000A positive power of ten moves the decimal point right by the exponent count.
Write 1.4 × 10^7 in standard notation.
14000000A positive power of ten moves the decimal point right by the exponent count.
Write 1.5 × 10^8 in standard notation.
150000000A positive power of ten moves the decimal point right by the exponent count.
Write 1.6 × 10^2 in standard notation.
160A positive power of ten moves the decimal point right by the exponent count.
Write 1.7 × 10^3 in standard notation.
1700A positive power of ten moves the decimal point right by the exponent count.
Write 1.0 × 10^4 in standard notation.
10000A positive power of ten moves the decimal point right by the exponent count.
Write 1.1 × 10^5 in standard notation.
110000A positive power of ten moves the decimal point right by the exponent count.
Write 1.2 × 10^6 in standard notation.
1200000A positive power of ten moves the decimal point right by the exponent count.
Write 1.3 × 10^7 in standard notation.
13000000A positive power of ten moves the decimal point right by the exponent count.
Practice
Guided practice with hints
01Write 1.4 × 10^8 in standard notation.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: 140000000
02Write 1.5 × 10^2 in standard notation.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: 150
03Write 1.6 × 10^3 in standard notation.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: 1600
04Write 1.7 × 10^4 in standard notation.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: 17000
05Write 1.0 × 10^5 in standard notation.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: 100000
06Write 1.1 × 10^6 in standard notation.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: 1100000
Practice
Independent practice
01Write 1.2 × 10^7 in standard notation.
Answer: 12000000
02Write 1.3 × 10^8 in standard notation.
Answer: 130000000
03Write 1.4 × 10^2 in standard notation.
Answer: 140
04Write 1.5 × 10^3 in standard notation.
Answer: 1500
05Write 1.6 × 10^4 in standard notation.
Answer: 16000
06Write 1.7 × 10^5 in standard notation.
Answer: 170000
07Write 1.0 × 10^6 in standard notation.
Answer: 1000000
08Write 1.1 × 10^7 in standard notation.
Answer: 11000000
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
Write 1.2 × 10^8 in standard notation.
120000000
Write 1.3 × 10^2 in standard notation.
130
Write 1.4 × 10^3 in standard notation.
1400
Write 1.5 × 10^4 in standard notation.
15000
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 8 Algebra & Modeling Sprint
Solve use powers of ten and scientific notation problems and justify each result with a valid verification method.
Practice
Mastery check
01Write 1.6 × 10^5 in standard notation.
Answer: 160000
02Write 1.7 × 10^6 in standard notation.
Answer: 1700000
03Write 1.0 × 10^7 in standard notation.
Answer: 10000000
04Write 1.1 × 10^8 in standard notation.
Answer: 110000000
05Write 1.2 × 10^2 in standard notation.
Answer: 120
06Write 1.3 × 10^3 in standard notation.
Answer: 1300
07Write 1.4 × 10^4 in standard notation.
Answer: 14000
08Write 1.5 × 10^5 in standard notation.
Answer: 150000
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.