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1.2.3.1: Calculations Involving Scientific Notation (TO DO)

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    Learning Objectives
    • Perform calculations with numbers in scientific notation

    Outside of simply expressing numbers in a less cumbersome manner, writing numbers in scientific notation can also simplify calculations. Since we'll be mostly utilizing these numbers in multiplication and division, we'll focus on those types of calculations on this page. Before discussing these calculations, let's go ahead and review the rules for multiplying and dividing exponential expressions.

    Multiplying and Dividing Exponents

    When multiplying exponential expressions that have the same base number a with exponents x and y, the final answer will be an exponential expression with the same base a but with the exponents added together.

    ax • ay = ax+y

    In this example, 54 is being multiplied by 53. The resulting expression is 57 which has the value 78125.

    54 • 53 = 57 = 78125

    When using negative exponents, remember that adding a negative number is essentially the same as subtracting. For example:

    54 • 5-3 = 54+(-3) = 54-3 = 51 = 5

    When dividing exponential expressions with the same base, the final answer will be an exponential expression with the same base a but with the numerator exponent subtracted by the denominator exponent. When performing division, it's imperative that you order the terms where the numerator is written first followed by the denominator.

    ax ÷ ay = ax / ay  = ax+y

    In this example, we are dividing 54 by 53 and the end result is simply 51 or 5.

    54 ÷ 53 = 54-3 =  51 = 5

    Be careful when using negative exponents. Recall that subtracting from a negative number is simply just adding the positive value of that number. Thus, when you're subtracting a negative number, this is the same as adding the two values as if they were both positive.

    54 ÷ 5-3 = 54-(-3) =  54+3 = 57 = 78125

    Here are the steps for multiplying or dividing two numbers in scientific notation.

    1. Multiply/divide the coefficients of each term together.
    2. Multiply/divide the powers of 10 by adding/subtracting their exponents.
    3. Convert your answer to proper scientific notation if necessary.
      • If coefficient >10, move the decimal x number of spaces to the left until it is one place to the right of the first non-zero number. Increase the exponent value by x
      • If coefficient <10, move the decimal x number of spaces to the right until it is one place to the right of the first non-zero number. Decrease the exponent value by x

    Here is an example of multiplying two numbers written in scientific notation together.

    Example \(\PageIndex{1}\)

    Multiply (3.25 x 10-3)(4.23 x 105)

    Solution
    1. (3.25)(4.23) = 13.7    Rounded due to significant figures
    2. (10-3)(105) = 10-3+5 = 102    
    3. 13.7 x 102
      • 13.7 is > 10. Move decimal one space to the left for 1.37. Add one to exponent power for 103

    Answer: 1.37 x 103    Proper way to express scientific notation is with decimal directly to the right of first non-zero. 

     

    Now let's look at dividing two numbers in scientific notation. Note that the order in which the terms are divided does matter.

    Example \(\PageIndex{1}\)

    Divide (3.25 x 10-3) ÷ (4.23 x 105)

    Solution
    1. (3.25) ÷ (4.23) = 0.768    Rounded due to significant figures
    2. (10-3) ÷ (105) = 10-3-5 = 10-8    
    3. 0.768 x 10-8
      • 0.768 is < 10. Move decimal one space to the right for 7.68. Subtract one from exponent power for 10-9

    Answer: 7.68 x 10-9    Proper way to express scientific notation is with decimal directly to the right of first non-zero. 

    Now that we're familiar with the calculations. Let's go ahead and solve a problem that contains numbers where it would be very useful to express in scientific notation.

    At its closest, the planet Neptune is 4,300,000,000 kilometers away from Earth. A group of astronauts from Earth want to make it to Neptune in 20,000 days. If they travel the same number of kilometers each day, how many kilometers will they travel each day? Convert both numbers to scientific notation before solving.

    First, convert both numbers to scientific notation.

    4,300,000,000=4.3×109

    20,000=2×104

    Next, notice that the astronauts want to travel the same distance each day. You will need to divide the total distance by the number of days to find the distance they will need to travel each day.

    (4.3×109)÷(2×104)

    Now, divide the decimal numbers using what you have learned about decimal division.

    4.3÷2=2.15

    Next, divide the powers of 10 by subtracting their exponents. 

    109 ÷104=109-4=105

    Now, combine the results.

    (4.3×109 )÷(2×104 )=(4.3÷2)×(109÷104)=2.15×105

    The answer is that the astronauts will need to travel 2.15×105 or 215,000 kilometers per day. They will be traveling for almost 55 years!

    Concept Review Exercises

    1. (3×104) × (2×103)

    2. (5.2 x 10-2) x (1.5 x 105)

    3. (9 × 108) ÷ (3×104)

    4. (2.4 x 10-6) ÷ (8 x 10-2)

    5. (6.02 x 10-2) ÷ (1.1 x 106) 

    6. (4.500 x 105) ÷ (2.0 x 103) 

    7. (9.21 x 10-1) ÷ (3.00 x 102) 

    8. (7.300 x 106) x (2.1 x 10-4) ÷ (5.00 x 102)

    Answers

    1. 6 x 107

    2. 7.8 x 103

    3. 3 x 104

    4. 3.0 x 10-5

    5. 6.6 x 104

    6. 2.3 x 102 = 230 (often just easier to express as a decimal)

    7. 3.07 x 10-3

    8. 3.1 x 100 = 3.1

    Key Takeaways

    • Multiplying and dividing numbers in scientific notation is done by multiplying/dividing the coefficients and the exponential expression separately and then rewriting back in desired format

    Attributions

    1.2.3.1: Calculations Involving Scientific Notation (TO DO) is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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