Learn on PengiBig Ideas Math, Course 2, AcceleratedChapter 6: Exponents and Scientific Notation

Lesson 7: Operations in Scientific Notation

In this Grade 7 lesson from Big Ideas Math Course 2 Accelerated, students learn how to add, subtract, and multiply numbers written in scientific notation by applying the Distributive Property and aligning powers of 10. Activities guide students to rewrite numbers so that exponents match before combining coefficients, a key step when the powers of 10 differ. This lesson builds directly on prior knowledge of scientific notation and prepares students for the operations with very large and very small quantities required by standards 8.EE.3 and 8.EE.4.

Section 1

Add Numbers in Scientific Notation

Property

To add numbers in scientific notation, first ensure they have the same power of 10. Then, add the decimal factors and keep the common power of 10. The general rule is (a×10n)+(b×10n)=(a+b)×10n(a \times 10^n) + (b \times 10^n) = (a + b) \times 10^n.

Examples

  • Same powers: (4.2×105)+(3.5×105)=(4.2+3.5)×105=7.7×105(4.2 \times 10^5) + (3.5 \times 10^5) = (4.2 + 3.5) \times 10^5 = 7.7 \times 10^5
  • Different powers: (6.1×103)+(2.5×104)=(0.61×104)+(2.5×104)=(0.61+2.5)×104=3.11×104(6.1 \times 10^3) + (2.5 \times 10^4) = (0.61 \times 10^4) + (2.5 \times 10^4) = (0.61 + 2.5) \times 10^4 = 3.11 \times 10^4

Explanation

When adding numbers in scientific notation, the exponents must be the same. If they are already the same, you can simply add the decimal parts and keep the common power of 10. If the exponents are different, you must first rewrite one of the numbers so that its exponent matches the other. Finally, ensure your answer is written in proper scientific notation.

Section 2

Subtract in Scientific Notation

Property

To subtract numbers in scientific notation, the powers of 10 must be the same. The subtraction is performed using the distributive property: (a×10n)−(b×10n)=(a−b)×10n(a \times 10^n) - (b \times 10^n) = (a - b) \times 10^n. If the powers are different, first rewrite one of the numbers so the exponents match.

Examples

  • Same powers: (8.5×104)−(2.1×104)=(8.5−2.1)×104=6.4×104(8.5 \times 10^4) - (2.1 \times 10^4) = (8.5 - 2.1) \times 10^4 = 6.4 \times 10^4
  • Different powers: (7.2×105)−(4.1×103)=(7.2×105)−(0.041×105)=(7.2−0.041)×105=7.159×105(7.2 \times 10^5) - (4.1 \times 10^3) = (7.2 \times 10^5) - (0.041 \times 10^5) = (7.2 - 0.041) \times 10^5 = 7.159 \times 10^5

Explanation

When subtracting numbers in scientific notation, you must ensure the exponents on the powers of 10 are identical. If they are, simply subtract the decimal factors and keep the same power of 10. If the exponents are different, you must adjust one of the numbers by moving its decimal point and changing its exponent until the exponents match. After matching the exponents, you can proceed with the subtraction.

Section 3

Multiply and Divide in Scientific Notation

Property

To multiply and divide numbers in scientific notation, group the coefficients together and group the powers of 10 together, then use the Properties of Exponents.

  • Multiplication: Multiply the decimal coefficients and ADD the exponents.
(a×10m)(b×10n)=(a⋅b)×10m+n(a \times 10^m)(b \times 10^n) = (a \cdot b) \times 10^{m+n}
  • Division: Divide the decimal coefficients and SUBTRACT the exponents.
a×10mb×10n=(ab)×10m−n\frac{a \times 10^m}{b \times 10^n} = \left(\frac{a}{b}\right) \times 10^{m-n}

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Add Numbers in Scientific Notation

Property

To add numbers in scientific notation, first ensure they have the same power of 10. Then, add the decimal factors and keep the common power of 10. The general rule is (a×10n)+(b×10n)=(a+b)×10n(a \times 10^n) + (b \times 10^n) = (a + b) \times 10^n.

Examples

  • Same powers: (4.2×105)+(3.5×105)=(4.2+3.5)×105=7.7×105(4.2 \times 10^5) + (3.5 \times 10^5) = (4.2 + 3.5) \times 10^5 = 7.7 \times 10^5
  • Different powers: (6.1×103)+(2.5×104)=(0.61×104)+(2.5×104)=(0.61+2.5)×104=3.11×104(6.1 \times 10^3) + (2.5 \times 10^4) = (0.61 \times 10^4) + (2.5 \times 10^4) = (0.61 + 2.5) \times 10^4 = 3.11 \times 10^4

Explanation

When adding numbers in scientific notation, the exponents must be the same. If they are already the same, you can simply add the decimal parts and keep the common power of 10. If the exponents are different, you must first rewrite one of the numbers so that its exponent matches the other. Finally, ensure your answer is written in proper scientific notation.

Section 2

Subtract in Scientific Notation

Property

To subtract numbers in scientific notation, the powers of 10 must be the same. The subtraction is performed using the distributive property: (a×10n)−(b×10n)=(a−b)×10n(a \times 10^n) - (b \times 10^n) = (a - b) \times 10^n. If the powers are different, first rewrite one of the numbers so the exponents match.

Examples

  • Same powers: (8.5×104)−(2.1×104)=(8.5−2.1)×104=6.4×104(8.5 \times 10^4) - (2.1 \times 10^4) = (8.5 - 2.1) \times 10^4 = 6.4 \times 10^4
  • Different powers: (7.2×105)−(4.1×103)=(7.2×105)−(0.041×105)=(7.2−0.041)×105=7.159×105(7.2 \times 10^5) - (4.1 \times 10^3) = (7.2 \times 10^5) - (0.041 \times 10^5) = (7.2 - 0.041) \times 10^5 = 7.159 \times 10^5

Explanation

When subtracting numbers in scientific notation, you must ensure the exponents on the powers of 10 are identical. If they are, simply subtract the decimal factors and keep the same power of 10. If the exponents are different, you must adjust one of the numbers by moving its decimal point and changing its exponent until the exponents match. After matching the exponents, you can proceed with the subtraction.

Section 3

Multiply and Divide in Scientific Notation

Property

To multiply and divide numbers in scientific notation, group the coefficients together and group the powers of 10 together, then use the Properties of Exponents.

  • Multiplication: Multiply the decimal coefficients and ADD the exponents.
(a×10m)(b×10n)=(a⋅b)×10m+n(a \times 10^m)(b \times 10^n) = (a \cdot b) \times 10^{m+n}
  • Division: Divide the decimal coefficients and SUBTRACT the exponents.
a×10mb×10n=(ab)×10m−n\frac{a \times 10^m}{b \times 10^n} = \left(\frac{a}{b}\right) \times 10^{m-n}