Learn on PengiBig Ideas Math, Course 3Chapter 10: Exponents and Scientific Notation

Lesson 7: Operations in Scientific Notation

In this Grade 8 lesson from Big Ideas Math Course 3, students learn how to add, subtract, multiply, and divide numbers written in scientific notation, including cases where the powers of 10 are the same or different. Key skills include applying the Distributive Property to combine like powers of 10, rewriting numbers to match exponents before adding or subtracting, and using the Product of Powers Property when multiplying. The lesson aligns with Common Core standards 8.EE.3 and 8.EE.4 and builds toward real-world estimation problems using scientific notation.

Section 1

Add Numbers in Scientific Notation with Same Powers

Property

When adding numbers in scientific notation with the same power of 10, use the distributive property:

(a×10n)+(b×10n)=(a+b)×10n(a \times 10^n) + (b \times 10^n) = (a + b) \times 10^n

Examples

Section 2

Subtract Numbers in Scientific Notation

Property

When subtracting numbers in scientific notation with the same power of 10, subtract the coefficients and keep the same power: (a×10n)−(b×10n)=(a−b)×10n(a \times 10^n) - (b \times 10^n) = (a - b) \times 10^n

Examples

Section 3

Add Numbers in Scientific Notation with Different Powers

Property

To add numbers in scientific notation with different powers of 10, rewrite one number so both have the same power of 10, then add the coefficients: (a×10m)+(b×10n)=(a′×10k)+(b′×10k)=(a′+b′)×10k(a \times 10^m) + (b \times 10^n) = (a' \times 10^k) + (b' \times 10^k) = (a' + b') \times 10^k

Examples

Section 4

Subtract Numbers in Scientific Notation with Different Powers

Property

To subtract numbers in scientific notation with different powers of 10, first rewrite one number so both have the same power of 10, then subtract the coefficients: (a×10m)−(b×10n)=(a×10m−n−b)×10n(a \times 10^m) - (b \times 10^n) = (a \times 10^{m-n} - b) \times 10^n when m>nm > n, or (a−b×10n−m)×10m(a - b \times 10^{n-m}) \times 10^m when n>mn > m.

Examples

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Add Numbers in Scientific Notation with Same Powers

Property

When adding numbers in scientific notation with the same power of 10, use the distributive property:

(a×10n)+(b×10n)=(a+b)×10n(a \times 10^n) + (b \times 10^n) = (a + b) \times 10^n

Examples

Section 2

Subtract Numbers in Scientific Notation

Property

When subtracting numbers in scientific notation with the same power of 10, subtract the coefficients and keep the same power: (a×10n)−(b×10n)=(a−b)×10n(a \times 10^n) - (b \times 10^n) = (a - b) \times 10^n

Examples

Section 3

Add Numbers in Scientific Notation with Different Powers

Property

To add numbers in scientific notation with different powers of 10, rewrite one number so both have the same power of 10, then add the coefficients: (a×10m)+(b×10n)=(a′×10k)+(b′×10k)=(a′+b′)×10k(a \times 10^m) + (b \times 10^n) = (a' \times 10^k) + (b' \times 10^k) = (a' + b') \times 10^k

Examples

Section 4

Subtract Numbers in Scientific Notation with Different Powers

Property

To subtract numbers in scientific notation with different powers of 10, first rewrite one number so both have the same power of 10, then subtract the coefficients: (a×10m)−(b×10n)=(a×10m−n−b)×10n(a \times 10^m) - (b \times 10^n) = (a \times 10^{m-n} - b) \times 10^n when m>nm > n, or (a−b×10n−m)×10m(a - b \times 10^{n-m}) \times 10^m when n>mn > m.

Examples