Learn on PengiReveal Math, Course 3Module 1: Exponents and Scientific Notation

Lesson 1-6: Compute with Scientific Notation

In this Grade 8 lesson from Reveal Math, Course 3 (Module 1: Exponents and Scientific Notation), students learn how to multiply, divide, add, and subtract numbers written in scientific notation. The lesson applies the Product of Powers Property, Quotient of Powers Property, and Distributive Property to solve real-world problems involving very large and very small numbers. Students also practice converting results into proper scientific notation form.

Section 1

Adjust Exponents when Shifting Decimals

Property

When adjusting a coefficient to proper scientific notation (1≤a<101 \leq a < 10), you must balance the decimal shift by changing the exponent:

  • Move decimal Left: Add to the exponent (+k+k)
  • Move decimal Right: Subtract from the exponent (−k-k)
c×10n=(c÷10k)×10n+kc \times 10^n = (c \div 10^k) \times 10^{n+k}
c×10n=(c×10k)×10n−kc \times 10^n = (c \times 10^k) \times 10^{n-k}

Section 2

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}

Section 3

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.

Book overview

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Module 1: Exponents and Scientific Notation

  1. Lesson 1

    Lesson 1-1: Powers and Exponents

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  2. Lesson 2

    Lesson 1-2: Multiply and Divide Monomials

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  3. Lesson 3

    Lesson 1-3: Powers of Monomials

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  4. Lesson 4

    Lesson 1-4: Zero and Negative Exponents

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  5. Lesson 5

    Lesson 1-5: Scientific Notation

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  6. Lesson 6Current

    Lesson 1-6: Compute with Scientific Notation

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Lesson overview

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Section 1

Adjust Exponents when Shifting Decimals

Property

When adjusting a coefficient to proper scientific notation (1≤a<101 \leq a < 10), you must balance the decimal shift by changing the exponent:

  • Move decimal Left: Add to the exponent (+k+k)
  • Move decimal Right: Subtract from the exponent (−k-k)
c×10n=(c÷10k)×10n+kc \times 10^n = (c \div 10^k) \times 10^{n+k}
c×10n=(c×10k)×10n−kc \times 10^n = (c \times 10^k) \times 10^{n-k}

Section 2

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}

Section 3

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.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Module 1: Exponents and Scientific Notation

  1. Lesson 1

    Lesson 1-1: Powers and Exponents

    LearnPractice
  2. Lesson 2

    Lesson 1-2: Multiply and Divide Monomials

    LearnPractice
  3. Lesson 3

    Lesson 1-3: Powers of Monomials

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  4. Lesson 4

    Lesson 1-4: Zero and Negative Exponents

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  5. Lesson 5

    Lesson 1-5: Scientific Notation

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  6. Lesson 6Current

    Lesson 1-6: Compute with Scientific Notation

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