How to Multiply Numbers in Scientific Notation?

Multiplying numbers in scientific notation might seem tricky at first, but it’s actually quite straightforward once you get the hang of it. Let’s break it down step by step.

What is Scientific Notation?

Scientific notation is a way to express very large or very small numbers conveniently. It has the form:
$a times 10^n$
where:

  • $a$ is a number greater than or equal to 1 and less than 10 (called the coefficient).
  • $n$ is an integer (called the exponent).

Steps to Multiply Numbers in Scientific Notation

To multiply numbers in scientific notation, follow these simple steps:

  1. Multiply the Coefficients
    First, multiply the coefficients (the $a$ values) together. For example, if you have $(2 times 10^3)$ and $(3 times 10^4)$, you would multiply 2 and 3:

    $2 times 3 = 6$

  1. Add the Exponents
    Next, add the exponents (the $n$ values) together. Using the same example, add 3 and 4:

    $3 + 4 = 7$

  1. Combine the Results
    Combine the product of the coefficients with the sum of the exponents to get the final result in scientific notation. So, $(2 times 10^3) times (3 times 10^4)$ becomes:

    $6 times 10^7$

Example Problem

Let’s look at a more detailed example to make sure you understand.

Example: Multiply $(4 times 10^2)$ by $(1.5 times 10^5)$

  1. Multiply the Coefficients:

$4 times 1.5 = 6$

  1. Add the Exponents:

$2 + 5 = 7$

  1. Combine the Results:

$(4 times 10^2) times (1.5 times 10^5) = 6 times 10^7$

What if the Coefficient is Not Between 1 and 10?

Sometimes, multiplying the coefficients gives you a number that isn’t between 1 and 10. In such cases, you’ll need to adjust the coefficient and exponent accordingly.

Example: Multiply $(5 times 10^3)$ by $(4 times 10^2)$

  1. Multiply the Coefficients:

$5 times 4 = 20$

  1. Add the Exponents:

$3 + 2 = 5$

  1. Adjust the Result: Since 20 is not between 1 and 10, you need to convert it:

$20 = 2 times 10^1$

  1. Combine and Adjust:

$20 times 10^5 = (2 times 10^1) times 10^5 = 2 times 10^{1+5} = 2 times 10^6$

So, $(5 times 10^3) times (4 times 10^2) = 2 times 10^6$

Conclusion

Multiplying numbers in scientific notation involves three simple steps: multiply the coefficients, add the exponents, and combine the results. If necessary, adjust the final coefficient to ensure it falls between 1 and 10. With a bit of practice, you’ll find it becomes second nature!

Citations

  1. 1. Khan Academy – Multiplying numbers in scientific notation
  2. 2. Math is Fun – Scientific Notation
  3. 3. Purplemath – Multiplying Scientific Notation

Related

(2) O3 + H → O2 + OH k2 = 1.78×10^-11 cm^3 s^-1 (3) O + OH → O2 + H k3 = 4.40×10^-11 cm^3 s^-1 (5) O + HO2 → O2 + OH k5 = 3.50×10^-11 cm^3 s^-1 (6) H + HO2 → O2 + H2 k6 = 5.40×10^-12 cm^3 s^-1 (9) OH + HO2 → O2 + H2O2 k9 = 4.00×10^-11 cm^3 s^-1 (10) HO2 + HO2 → O2 + H2O2 k10 = 2.50×10^-12 cm s^-1 (11) O + O2 + M → O3 + M k11 = 1.05×10^-34 cm^6 s^-1 (14) H + O2 + M → HO2 + M k14 = 8.08×10^-32 cm^6 s^-1 (15) H + H + M → H2O + M k15 = 3.31×10^-27 cm^6 s^-1 (16) O2 + hv → 2 O k16 = (1.26×10^-8 s^-1) φ (17) H2O + hv → H + OH k17 = (3.4×10^-6 s^-1) φ (18) O3 + hv → O2 + O k18 = (7.10×10^-5 s^-1) φ

Table 1 Reactions, rate constants and activation energies used in the model* No. Reaction kopt (M⁻¹ s⁻¹) 1 OH + H₂ → H + H₂O 3.74 x 10⁷ 2 OH + HO₂ → HO₂ + OH⁻ 5 x 10⁹ 3 OH + H₂O₂ → HO₂ + H₂O 3.8 x 10⁷ 4 OH + O₂ → O₂ + OH 9.96 x 10⁹ 5 OH + HO₂ → O₂ + H₂O 7.1 x 10⁹ 6 OH + OH → H₂O₂ 5.3 x 10⁹ 7 OH + e⁻aq → OH⁻ 3 x 10¹⁰ 8 H + O₂ → HO₂ 2.0 x 10¹⁰ 9 H + HO₂ → H₂O₂ 2.0 x 10¹⁰ 10 H + H₂O₂ → OH + H₂O 3.44 x 10⁷ 11 H + OH → H₂O 1.4 x 10¹⁰ 12 H + H → H₂ 1.94 x 10¹⁰ 13 e⁻aq + O₂ → O₂⁻ 1.9 x 10¹⁰ 14 e⁻aq + O₂ → HO₂⁻ + OH⁻ 1.3 x 10¹⁰ 15 e⁻aq + HO₂ 2.0 x 10¹⁰ 16 e⁻aq + H₂O₂ 1.1 x 10¹⁰ 17 e⁻aq + HO₂ → OH + OH⁻ 1.3 x 10¹⁰ 18 e⁻aq + H⁺ → H 2.3 x 10¹⁰ 19 e⁻aq + e⁻aq → H₂ + OH⁻ + OH⁻ 2.5 x 10⁹ 20 HO₂ + O₂ → O₂ + HO₂ 1.3 x 10⁹ 21 HO₂ + HO₂ → O₂ + H₂O₂ 8.3 x 10⁵ 22 HO₂ + HO₂ → O₂ + OH + H₂O 3.7 23 HO₂ + HO₂ → O₂ + O₂ + OH + H₂O 7 x 10⁵ s⁻¹ 24 H⁺ + O₂⁻ → HO₂ 4.5 x 10¹⁰ 25 H⁺ + O₂⁻ → O₂ 2.0 x 10¹⁰ 26 H⁺ + OH⁻ 1.4 x 10¹¹ 27 H⁺ + HO₂⁻ 2 x 10¹⁰ 28 H₂O₂ → HO₂ + H⁺ + OH⁻ 2.5 x 10⁻⁵ s⁻¹ 29 H₂O₂ → H⁺ + OH⁻ 1.4 x 10⁻⁷ s⁻¹ 30 O₂ + O₂ → O₂ + HO₂ + OH⁻ 0.3 31 O₂ + H₂O₂ → O₂ + OH + OH 16 32

(2) O3 + H → O2 + OH k2 = 1.78×10^-11 cm^3 s^-1 (3) O + OH → O2 + H k3 = 4.40×10^-11 cm^3 s^-1 (5) O + HO2 → O2 + OH k5 = 3.50×10^-11 cm^3 s^-1 (6) H2O + O → 2 OH k6 = 5.40×10^-12 cm^3 s^-1 (9) OH + HO2 → O2 + H2O k9 = 4.00×10^-11 cm^3 s^-1 (10) HO2 + HO2 → O2 + H2O2 k10 = 2.50×10^-12 cm s^-1 (11) O + O2 + M → O3 + M k11 = 1.05×10^-34 cm^6 s^-1 (14) H + O2 + M → HO2 + M k14 = 8.08×10^-32 cm^6 s^-1 (15) OH + H + M → H2O + M k15 = 3.31×10^-27 cm^6 s^-1 (16) O2 + hv → 2 O k16 = (1.26×10^-8 s^-1) φ (17) H2O + hv → H + OH k17 = (3.4×10^-6 s^-1) φ (18) O3 + hv → O2 + O k18 = (7.10×10^-8 s^-1) φ